Search: a013655 -id:a013655
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A274285
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Numbers that are a product of distinct numbers in A013655.
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+20
2
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2, 5, 7, 10, 12, 14, 19, 24, 31, 35, 38, 50, 60, 62, 70, 81, 84, 95, 100, 120, 131, 133, 155, 162, 168, 190, 212, 217, 228, 250, 262, 266, 310, 343, 350, 372, 405, 420, 424, 434, 456, 500, 555, 567, 589, 600, 655, 665, 686, 700, 744, 810, 840, 898, 917, 950
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OFFSET
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1,1
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COMMENTS
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See the Comment on distinct-product sequences in A160009.
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LINKS
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EXAMPLE
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10 = 2*5, 120 = 2*5*12.
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MATHEMATICA
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f[1] = 2; f[2] = 5; z = 33; f[n_] := f[n - 1] + f[n - 2]; f = Table[f[n], {n, 1, z}]; f
s = {1}; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s1 = Rest[s]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A035513
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Wythoff array read by antidiagonals.
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+10
172
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1, 2, 4, 3, 7, 6, 5, 11, 10, 9, 8, 18, 16, 15, 12, 13, 29, 26, 24, 20, 14, 21, 47, 42, 39, 32, 23, 17, 34, 76, 68, 63, 52, 37, 28, 19, 55, 123, 110, 102, 84, 60, 45, 31, 22, 89, 199, 178, 165, 136, 97, 73, 50, 36, 25, 144, 322, 288, 267, 220, 157, 118, 81, 58, 41, 27, 233, 521
(list;
table;
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OFFSET
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1,2
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COMMENTS
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T(0,0)=1, T(0,1)=2,...; y^2-x^2-xy<y if and only if there exist (i,j) with x=T(i,2j) and y=T(i,2j+1). - Claude Lenormand (claude.lenormand(AT)free.fr), Mar 17 2001
The Wythoff array W consists of all the Wythoff pairs (x(n),y(n)), where x=A000201 and y=A001950, so that W contains every positive integer exactly once. The differences T(i,2j+1)-T(i,2j) form the Wythoff difference array, A080164, which also contains every positive integer exactly once. - Clark Kimberling, Feb 08 2003
For n>2 the determinant of any n X n contiguous subarray of A035513 (as a square array) is 0. - Gerald McGarvey, Sep 18 2004
Except for initial terms in some cases:
(Row 8) = A013655 (sum of Fibonacci and Lucas numbers)
(Column 1) = A003622 = AA Wythoff sequence
(Column 2) = A035336 = BA Wythoff sequence
(Column 3) = A035337 = ABA Wythoff sequence
(Column 4) = A035338 = BBA Wythoff sequence
(Column 5) = A035339 = ABBA Wythoff sequence
(Column 6) = A035340 = BBBA Wythoff sequence
The Wythoff array is the dispersion of the sequence given by floor(n*x+x-1), where x=(golden ratio). See A191426 for a discussion of dispersions. - Clark Kimberling, Jun 03 2011
If u and v are finite sets of numbers in a row of the Wythoff array such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. See A160009 (row 1 products), A274286 (row 2), A274287 (row 3), A274288 (row 4). - Clark Kimberling, Jun 17 2016
All columns of the Wythoff array are compound Wythoff sequences. This follows from the main theorem in the 1972 paper by Carlitz, Scoville and Hoggatt. For an explicit expression see Theorem 10 in Kimberling's paper from 2008 in JIS. - Michel Dekking, Aug 31 2017
The Wythoff array can be viewed as an infinite graph over the set of nonnegative integers, built as follows: start with an empty graph; for all n = 0, 1, ..., create an edge between n and the sum of the degrees of all i < n. Finally, remove vertex 0. In the resulting graph, the connected components are chains and correspond to the rows of the Wythoff array. - Luc Rousseau, Sep 28 2017
Suppose that h < k are consecutive terms in a row of the Wythoff array. If k is in an even numbered column, then h = floor(k/tau); otherwise, h = -1 + floor(k/tau). - Clark Kimberling, Mar 05 2020
For k > = 0, column k shows the numbers m having F(k+1) as least term in the Zeckendorf representation of m. For n >= 1, let r(n,k) be the number of terms in column k that are <= n. Then n/r(n,k) = n/(F(k+1)*tau + F(k)*(n-1)), by Bottomley's formula, so that the limiting ratio is 1/(F(k+1)*tau + F(k)). Summing over all k gives Sum_{k>=0} 1/(F(k+1)*tau + F(k)) = 1. Thus, in the limiting sense:
38.19...% of the numbers m have least term 1;
23.60...% have least term 2;
14.58...% have least term 3;
9.01...% have least term 5, etc. (End)
Named after the Dutch mathematician Willem Abraham Wythoff (1865-1939). - Amiram Eldar, Jun 11 2021
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REFERENCES
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John H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.
Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
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LINKS
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L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Fibonacci representations, Fib. Quart., Vol. 10, No. 1 (1972), pp. 1-28.
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FORMULA
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T(n,-1) = n-1. T(n,0) = floor(n*tau). T(n,k) = T(n,k-1) + T(n,k-2) for k>=1. - R. J. Mathar, Sep 03 2016
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EXAMPLE
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The Wythoff array begins:
1 2 3 5 8 13 21 34 55 89 144 ...
4 7 11 18 29 47 76 123 199 322 521 ...
6 10 16 26 42 68 110 178 288 466 754 ...
9 15 24 39 63 102 165 267 432 699 1131 ...
12 20 32 52 84 136 220 356 576 932 1508 ...
14 23 37 60 97 157 254 411 665 1076 1741 ...
17 28 45 73 118 191 309 500 809 1309 2118 ...
19 31 50 81 131 212 343 555 898 1453 2351 ...
22 36 58 94 152 246 398 644 1042 1686 2728 ...
25 41 66 107 173 280 453 733 1186 1919 3105 ...
27 44 71 115 186 301 487 788 1275 2063 3338 ...
...
The extended Wythoff array has two extra columns, giving the row number n and A000201(n), separated from the main array by a vertical bar:
0 1 | 1 2 3 5 8 13 21 34 55 89 144 ...
1 3 | 4 7 11 18 29 47 76 123 199 322 521 ...
2 4 | 6 10 16 26 42 68 110 178 288 466 754 ...
3 6 | 9 15 24 39 63 102 165 267 432 699 1131 ...
4 8 | 12 20 32 52 84 136 220 356 576 932 1508 ...
5 9 | 14 23 37 60 97 157 254 411 665 1076 1741 ...
6 11 | 17 28 45 73 118 191 309 500 809 1309 2118 ...
7 12 | 19 31 50 81 131 212 343 555 898 1453 2351 ...
8 14 | 22 36 58 94 152 246 398 644 1042 1686 2728 ...
9 16 | 25 41 66 107 173 280 453 733 1186 1919 3105 ...
10 17 | 27 44 71 115 186 301 487 788 1275 2063 3338 ...
11 19 | 30 49 79 ...
12 21 | 33 54 87 ...
13 22 | 35 57 92 ...
14 24 | 38 62 ...
15 25 | 40 65 ...
16 27 | 43 70 ...
17 29 | 46 75 ...
18 30 | 48 78 ...
19 32 | 51 83 ...
20 33 | 53 86 ...
21 35 | 56 91 ...
22 37 | 59 96 ...
23 38 | 61 99 ...
24 40 | 64 ...
25 42 | 67 ...
26 43 | 69 ...
27 45 | 72 ...
28 46 | 74 ...
29 48 | 77 ...
30 50 | 80 ...
31 51 | 82 ...
32 53 | 85 ...
33 55 | 88 ...
34 56 | 90 ...
35 58 | 93 ...
36 59 | 95 ...
37 61 | 98 ...
38 63 | ...
...
Each row of the extended Wythoff array also satisfies the Fibonacci recurrence, and may be extended to the left using this recurrence backwards.
The Wythoff array appears to have the following relationship to the traditional Fibonacci rabbit breeding story, modified for simplicity to be a story of asexual reproduction.
Give each rabbit a number, 0 for the initial rabbit.
When a new round of rabbits is born, allocate consecutive numbers according to 2 rules (the opposite of many cultural rules for inheritance precedence): (1) newly born child of Rabbit 0 gets the next available number; (2) the descendants of a younger child of any given rabbit precede the descendants of an older child of the same rabbit.
Row n of the Wythoff array lists the children of Rabbit n (so Rabbit 0's children have the Fibonacci numbers: 1, 2, 3, 5, ...). The generation tree below shows rabbits 0 to 20. It is modified so that each round of births appears on a row.
0
:
,-------------------------:
: :
,---------------: 1
: : :
,--------: 2 ,---------:
: : : : :
,-----: 3 ,-----: ,-----: 4
: : : : : : : :
,--: 5 ,--: ,---: 6 ,---: 7 ,---:
: : : : : : : : : : : : :
,--: 8 ,--: ,--: 9 ,--: 10 ,--: ,--: 11 ,--: ,--: 12
: : : : : : : : : : : : : : : : : : : : :
: 13 : : 14 : 15 : : 16 : : 17 : 18 : : 19 : 20 :
The extended array's nontrivial extra column (A000201) gives the number that would have been allocated to the first child of Rabbit n, if Rabbit n (and only Rabbit n) had started breeding one round early.
(End)
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MAPLE
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W:= proc(n, k) Digits:= 100; (Matrix([n, floor((1+sqrt(5))/2* (n+1))]). Matrix([[0, 1], [1, 1]])^(k+1))[1, 2] end: seq(seq(W(n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Aug 18 2008
option remember;
if c = 1 then
else
end if;
end proc:
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MATHEMATICA
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W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; Table[ W[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten
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PROG
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(PARI) T(n, k)=(n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)
(Python)
from sympy import fibonacci as F, sqrt
import math
tau = (sqrt(5) + 1)/2
def T(n, k): return F(k + 1)*int(math.floor(n*tau)) + F(k)*(n - 1)
for n in range(1, 11): print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 23 2017
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CROSSREFS
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See comments above for more cross-references.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Comments about the extended Wythoff array added by N. J. A. Sloane, Mar 07 2016
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STATUS
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approved
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A000285
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a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2.
(Formerly M3246 N1309)
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+10
47
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1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505, 16276621, 26336126, 42612747, 68948873, 111561620, 180510493, 292072113, 472582606
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OFFSET
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0,2
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COMMENTS
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a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(4;n-1-k,k), n >= 1, with a(-1)=3. These are the sums over the SW-NE diagonals in P(4;n,k), the (4,1) Pascal triangle A093561. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums in the Pascal (1,3) triangle A095660.
In general, for a Fibonacci sequence beginning with 1,b we have a(n) = (2^(-1-n)*((1-sqrt(5))^n*(1+sqrt(5)-2b) + (1+sqrt(5))^n*(-1+sqrt(5)+2b)))/sqrt(5). In this case we have b=4. - Herbert Kociemba, Dec 18 2011
Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 5, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... - R. J. Mathar, Aug 10 2012
a(n) = number of independent vertex subsets (i.e., the Merrifield-Simmons index) of the tree obtained from the path tree P_{n-1} by attaching two pendant edges to one of its endpoints (n >= 2). Example: if n=3, then we have the star tree with edges ab, ac, ad; it has 9 independent vertex subsets: empty, a, b, c, d, bc, cd, bd, bcd.
For n >= 2, the number a(n-1) is the dimension of a commutative Hecke algebra of type D_n with independent parameters. See Theorem 1.4 and Corollary 1.5 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019
For n >= 1, a(n) is the number of edge covers of the tadpole graph T_{3,n-1} with T_{3,0} interpreted as just the cycle C_3. Example: If n=2, we have C_3 and P_1 joined by a bridge, which is just the triangle with a pendant, and this graph has 5 edge covers. In general, because of the path portion of the graph, the number of edge covers of T{3,n-1} satisfies the same recurrence as Fibonacci sequence and it starts with 4,5. - Feryal Alayont, Aug 27 2023
Eswarathasan (1978) called these numbers "pseudo-Fibonacci numbers", and proved that 1, 4, and 9 are the only squares in this sequence. If the recurrence is extended to negative indices, then there is only one more square, a(-9) = 81. Eswarathasan (1979) proved that none of the terms (even with negative indices) are twice a square. - Amiram Eldar, Mar 09 2024
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REFERENCES
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Richard E. Merrifield and Howard E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 131.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (1+3*x)/(1-x-x^2). - Simon Plouffe in his 1992 dissertation
a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + (3/2)* ((1+sqrt(5))^(n-1) - (1-sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). Offset 1. a(3)=5. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009
a(n) = 3*Fibonacci(n+2) - 2*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010
a(n) = Fibonacci(n) + Lucas(n+1), see Mathematica field. (End)
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EXAMPLE
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G.f. = 1 + 4*x + 5*x^2 + 9*x^3 + 14*x^4 + 23*x^5 + 37*x^6 + 60*x^7 + ...
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MAPLE
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with(combinat):a:=n->2*fibonacci(n)+fibonacci(n+2): seq(a(n), n=0..34);
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MATHEMATICA
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LinearRecurrence[{1, 1}, {1, 4}, 40] (* or *) Table[(3*LucasL[n]- Fibonacci[n])/2, {n, 40}] (* Harvey P. Dale, Jul 18 2011 *)
a[ n_]:= Fibonacci[n] + LucasL[n+1]; (* Michael Somos, May 28 2014 *)
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PROG
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(Haskell)
a000285 n = a000285_list !! n
a000285_list = 1 : 4 : zipWith (+) a000285_list (tail a000285_list)
(Maxima) a[0]:1$ a[1]:4$ a[n]:=a[n-1]+a[n-2]$ makelist(a[n], n, 0, 30); /* Martin Ettl, Oct 25 2012 */
(Magma) a0:=1; a1:=4; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013
(Sage) f=fibonacci; [f(n+2) +2*f(n) for n in (0..40)] # G. C. Greubel, Nov 08 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+2) +2*F(n) ); // G. C. Greubel, Nov 08 2019
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CROSSREFS
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Essentially the same as A104449, which only has A104449(0)=3 prefixed.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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A160009
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Numbers that are the product of distinct Fibonacci numbers.
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+10
26
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0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 30, 34, 39, 40, 42, 48, 55, 63, 65, 68, 78, 80, 89, 102, 104, 105, 110, 120, 126, 130, 144, 165, 168, 170, 178, 195, 204, 208, 210, 233, 240, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 377, 390, 432, 440, 442, 445
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OFFSET
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1,3
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COMMENTS
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Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.
It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. The same holds for many other 2nd order linear recurrence sequences with constant coefficients. In the following guide to related "distinct product sequences", W = Wythoff array, A035513:
base sequence distinct-product sequence
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LINKS
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MATHEMATICA
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s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s, Select[s*f[[i]], #<=f[[nn]]&]], {i, nn}]; s=Prepend[s, 0]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A001060
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a(n) = a(n-1) + a(n-2) with a(0)=2, a(1)=5. Sometimes called the Evangelist Sequence.
(Formerly M1338 N0512)
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+10
19
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2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, 16114, 26073, 42187, 68260, 110447, 178707, 289154, 467861, 757015, 1224876, 1981891, 3206767, 5188658, 8395425, 13584083, 21979508, 35563591, 57543099, 93106690, 150649789
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OFFSET
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0,1
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COMMENTS
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Used by the Sofia Gubaidulina and other composers. - Ian Stewart, Jun 07 2012
From a(2) on, sums of five consecutive Fibonacci numbers; the subset of primes is essentially in A153892. - R. J. Mathar, Mar 24 2010
Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (is this A001175?). - R. J. Mathar, Aug 10 2012
Also the number of independent vertex sets and vertex covers in the (n+1)-pan graph. - Eric W. Weisstein, Jun 30 2017
For n > 0, a(n) is the number of ways to tile the figure below with squares and dominoes (a strip of length n+1 that contains a vertical strip of height 3 in its second tile). For instance, a(4) is the number of ways to tile this figure (of length 5) with squares and dominoes.
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_|_|_______
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(End)
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REFERENCES
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R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Eric Weisstein's World of Mathematics, Pan Graph
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FORMULA
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a(n) = Fibonacci(n+4) - Fibonacci(n-1) for n >= 1. - Ian Stewart, Jun 07 2012
a(n) = Fibonacci(n) + 2*Fibonacci(n+2) = 5*Fibonacci(n) + 2*Fibonacci(n-1). The ratio r(n) := a(n+2)/a(n) satisfies the recurrence r(n+1) = (2*r(n) - 1)/(r(n) - 1). If M denotes the 2 X 2 matrix [2, -1; 1, -1] then [a(n+2), a(n)] = M^n[2, -1]. - Peter Bala, Dec 06 2013
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MAPLE
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with(combinat): a:= n-> 2*fibonacci(n)+fibonacci(n+3): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007
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MATHEMATICA
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CoefficientList[Series[(2+3x)/(1-x-x^2), {x, 0, 40}], x] (* Eric W. Weisstein, Sep 22 2017 *)
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PROG
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(Magma) I:=[2, 5]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jan 16 2012
(Magma) a0:=2; a1:=5; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..35]]; // Bruno Berselli, Feb 12 2013
(Sage) f=fibonacci; [f(n+4) - f(n-1) for n in (0..40)] # G. C. Greubel, Sep 19 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) - F(n-1) ); # G. C. Greubel, Sep 19 2019
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CROSSREFS
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Apart from initial term, same as A013655.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A104449
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Fibonacci sequence with initial values a(0) = 3 and a(1) = 1.
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+10
13
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3, 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505, 16276621, 26336126, 42612747, 68948873, 111561620
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OFFSET
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0,1
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COMMENTS
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The old name was: The Pibonacci numbers (a Fibonacci-type sequence): each term is the sum of the two previous terms.
The 6th row in the Wythoff array begins with the 6th term of the sequence (14, 23, 37, 60, 97, 157, ...). a(n) = f(n-3) + f(n+2) for the Fibonacci numbers f(n) = f(n-1) + f(n-2); f(0) = 0, f(1) = 1.
(a(2*k), a(2*k+1)) give for k >= 0 the proper positive solutions of one of two families (or classes) of solutions (x, y) of the indefinite binary quadratic form x^2 + x*y - y^2 of discriminant 5 representing 11. The other family of such solutions is given by (x2, y2) = (b(2*k), b(2*k+1)) with b = A013655. See the formula in terms of Chebyshev S polynomials S(n, 3) = A001906(n+1) below, which follows from the fundamental solution (3, 1) by applying positive powers of the automorphic matrix given in a comment in A013655. See also A089270 with the Alfred Brousseau link with D = 11. - Wolfdieter Lang, May 28 2019
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REFERENCES
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V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-2) with a(0) = 3, a(1) = 1.
a(n) = ( (3*sqrt(5)-1)*((1+sqrt(5))/2)^n + (3*sqrt(5)+1)*((1-sqrt(5) )/2)^n )/(2*sqrt(5)). - Bogart B. Strauss, Jul 19 2013
Bisection: a(2*k) = 4*S(k-1, 3) - 3*S(k-2, 3), a(2*k+1) = 2*S(k-1, 3) + S(k, 3) for k >= 0, with the Chebyshev S(n, 3) polynomials from A001906(n+1) for n >= -1. - Wolfdieter Lang, May 28 2019
a(3n + 4)/a(3n + 1) = continued fraction 4,4,4,...,4,9 (that's n 4's followed by a single 9). - Greg Dresden and Shaoxiong Yuan, Jul 16 2019
E.g.f.: (exp((1/2)*(1 - sqrt(5))*x)*(1 + 3*sqrt(5) + (- 1 + 3*sqrt(5))*exp(sqrt(5)*x)))/(2*sqrt(5)). - Stefano Spezia, Jul 18 2019
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MAPLE
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a:=n->3*fibonacci(n-1)+fibonacci(n): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007
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MATHEMATICA
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LinearRecurrence[{1, 1}, {3, 1}, 40] (* Harvey P. Dale, May 23 2014 *)
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PROG
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(Magma) [Fibonacci(n-1) + Lucas(n): n in [0..40]]; // G. C. Greubel, May 29 2019
(Sage) ((3-2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 29 2019
(GAP) a:=[3, 1];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # G. C. Greubel, May 29 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A119286
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Alternating sum of the fifth powers of the first n Fibonacci numbers.
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+10
9
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0, -1, 0, -32, 211, -2914, 29854, -341439, 3742662, -41692762, 461591613, -5122467836, 56794896388, -629924960005, 6985721085652, -77473909014348, 859194263419359, -9528629686028398, 105674040835291026, -1171943417651373875, 12997050199917354250, -144139501695851560726, 1598531543102764228825, -17727986584911448406232, 196606383515036414871336, -2180398207207766329269289
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OFFSET
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0,4
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COMMENTS
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Natural bilateral extension (brackets mark index 0): ..., 3402, 277, 34, 2, 1, 0, [0], -1, 0, -32, 211, -2914, 29854, ... This is A098531-reversed followed by A119286.
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LINKS
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FORMULA
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Let F(n) be the Fibonacci number A000045(n).
a(n) = Sum_{k=1..n} (-1)^k F(k)^5.
Closed form: a(n) = (-1)^n (1/275)(F(5n+1) + 2 F(5n+3)) - (1/10) F(3n+2) + (-1)^n (2/5) F(n-1) - 7/22; here F(5n+1) + 2 F(5n+3) = A001060(5n+1) = A013655(5n+2).
Recurrence: a(n) + 7 a(n-1) - 48 a(n-2) - 20 a(n-3) + 100 a(n-4) - 32 a(n-5) - 9 a(n-6) + a(n-7) = 0.
G.f.: A(x) = (-x - 7 x^2 + 16 x^3 + 7 x^4 - x^5)/(1 + 7 x - 48 x^2 - 20 x^3 + 100 x^4 - 32 x^5 - 9 x^6 + x^7) = -x(1 + 7 x - 16 x^2 - 7 x^3 + x^4)/((1 - x)(1 + x - x^2)(1 - 4 x - x^2)(1 + 11 x - x^2)).
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MATHEMATICA
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a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[k]^5, {k, 1, n} ], Sum[ -(-1)^k Fibonacci[ -k]^5, {k, 1, -n - 1} ] ]
LinearRecurrence[{-7, 48, 20, -100, 32, 9, -1}, {0, -1, 0, -32, 211, -2914, 29854}, 30] (* Harvey P. Dale, Jun 24 2018 *)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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A332938
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Indices of the primitive rows of the Wythoff array (A035513); see Comments.
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+10
8
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1, 2, 6, 7, 8, 10, 11, 12, 14, 17, 18, 20, 21, 23, 24, 26, 27, 30, 32, 33, 36, 37, 38, 39, 40, 42, 44, 46, 48, 49, 50, 53, 54, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 79, 80, 81, 84, 85, 86, 88, 90, 92, 94, 95, 98, 100, 101, 102, 104, 107
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OFFSET
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1,2
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COMMENTS
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In a row of the Wythoff array, either every two consecutive terms are relatively prime or else no two consecutive terms are relatively prime. In the first case, we call the row primitive; otherwise, the row is an integer multiple of a tail of a preceding row. Conjectures: the maximal number of consecutive primitive rows is 5, and the limiting proportion of primitive rows exists and is approximately 0.608.
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LINKS
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EXAMPLE
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The Wythoff array begins:
1 2 3 5 8 13 21 34 55 89 144 ...
4 7 11 18 29 47 76 123 199 322 521 ...
6 10 16 26 42 68 110 178 288 466 754 ...
9 15 24 39 63 102 165 267 432 699 1131 ...
12 20 32 52 84 136 220 356 576 932 1508 ...
14 23 37 60 97 157 254 411 665 1076 1741 ...
17 28 45 73 118 191 309 500 809 1309 2118 ...
19 31 50 81 131 212 343 555 898 1453 2351 ...
22 36 58 94 152 246 398 644 1042 1686 2728 ...
Row 1: A000045 (Fibonacci numbers, a primitive row)
Row 2: A000032 (Lucas numbers, primitive)
Row 3: 2 times a tail of row 1
Row 4: 3 times a tail of row 1
Row 5 4 times a tail of row 1
Row 6: essentially A000285, primitive
Row 7: essentially A022095, primitive
Row 8: essentially A013655, primitive
Row 9: 2 times a tail of row 2
Thus first five terms of (a(n)) are 1,2,6,7,8.
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MATHEMATICA
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W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; (* A035513 *)
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}] (* A332937 *)
Flatten[Position[t, 1]] (* A332938 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A100545
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Expansion of (7-2*x) / (1-3*x+x^2).
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+10
7
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7, 19, 50, 131, 343, 898, 2351, 6155, 16114, 42187, 110447, 289154, 757015, 1981891, 5188658, 13584083, 35563591, 93106690, 243756479, 638162747, 1670731762, 4374032539, 11451365855, 29980065026, 78488829223, 205486422643, 537970438706, 1408424893475, 3687304241719, 9653487831682, 25273159253327
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OFFSET
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0,1
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COMMENTS
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A Floretion integer sequence relating to Fibonacci numbers.
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LINKS
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FORMULA
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a(n-1) = 4*Fibonacci(2*n) + Fibonacci(2*n-1) + Fibonacci(2*n+1).
a(n) = 3*a(n-1) - a(n-2) with a(0)=7 and a(1)=19. - Philippe Deléham, Nov 16 2008
a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-17+7*sqrt(5)) + (3+sqrt(5))^n*(17+7*sqrt(5)))) / sqrt(5). - Colin Barker, Oct 14 2015
a(n) = Fibonacci(2*n+4) + Lucas(2*n+3).
E.g.f.: 2*exp(3*t/2)*(cosh(sqrt(5)*t/2) + (4/sqrt(5))*sinh(sqrt(5)*t/2)). (End)
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MAPLE
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F := proc(n) combinat[fibonacci](n) ; end: A100545 := proc(n) 4*F(2*(n+1)) + F(2*n+1)+F(2*n+3) ; end: for n from 0 to 30 do printf("%d, ", A100545(n)) ; od ; # R. J. Mathar, Oct 26 2006
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MATHEMATICA
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Table[Fibonacci[2*(n+2)] + LucasL[2*n+3], {n, 0, 30}] (* G. C. Greubel, Jan 17 2020 *)
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PROG
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(PARI) Vec((7-2*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 11 2015
(Magma) [Fibonacci(2*n+4) +Lucas(2*n+3): n in [0..30]]; // G. C. Greubel, Jan 17 2020
(Sage) [fibonacci(2*n+4) +lucas_number2(2*n+3, 1, -1) for n in (0..30)] # G. C. Greubel, Jan 17 2020
(GAP) List([0..30], n-> Fibonacci(2*n+4) +Lucas(1, -1, 2*n+3)[2] ); # G. C. Greubel, Jan 17 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A230871
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Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.
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+10
6
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0, 1, 1, 3, 2, 2, 4, 8, 3, 5, 3, 5, 7, 9, 11, 21, 5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55, 8, 12, 10, 18, 12, 16, 18, 34, 8, 12, 10, 18, 12, 16, 18, 34, 18, 26, 24, 44, 22, 30, 32, 60, 30, 46, 36, 64, 50, 66, 76, 144, 13, 19, 17, 31, 17, 23
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OFFSET
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0,4
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COMMENTS
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The rule for constructing the tree is the following:
.....x
.....|
.....y
..../ \
..y+x..3y-x
and the tree begins like this:
.........0......
.........|......
.........1......
......./ \....
......1.....3....
...../ \.../ \...
....2...2.4...8..
and so on.
Column 1 : 0, 1, 1, 2, 3, 5, 8, ... = A000045 (Fibonacci numbers).
Column 2 : 3, 2, 5, 7, 12, 19, 31, ... = A013655.
Column 3 : 4, 3, 7, 10, 17, 27, 44, ... = A022120.
Column 4 : 8, 5, 13, 18, 31, 49, 80, ... = A022138.
Column 5 : 7, 5, 12, 17, 29, 46, 75, ... = A022137.
Column 6 : 9, 7, 16, 23, 39, 62, 101, ... = A190995.
Column 7 : 11, 7, 18, 25, 43, 68, 111, ... = A206419.
Column 8 : 21, 13, 34, 47, 81, 128, 209, ... = ?
Column 9 : 11, 8, 19, 27, 46, 73, 119, ... = A206420.
The lengths of the rows are 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ... = A011782 .
The final numbers in the rows are 0, 1, 3, 8, 21, 55, 144, ... = A001906.
The middle numbers in the rows are 1, 2, 5, 13, 34, 89, ... = A001519 .
Row sums for n>=1: 1, 4, 16, 64, 256, 1024, ... = 4^(n-1).
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LINKS
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EXAMPLE
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The successive rows are:
0
1
1, 3
2, 2, 4, 8
3, 5, 3, 5, 7, 9, 11, 21
5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55
...
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MAPLE
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T:= proc(n, k) T(n, k):= `if`(k=1 and n<2, n, (d->(1+2*d)*
T(n-1, r)+(1-2*d)*T(n-2, iquo(r+1, 2)))(irem(k+1, 2, 'r')))
end:
seq(seq(T(n, k), k=1..max(1, 2^(n-1))), n=0..7); # Alois P. Heinz, Nov 07 2013
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[k==1 && n<2, n, Function[d, r = Quotient[k+1, 2]; (1+2d) T[n-1, r] + (1-2d) T[n-2, Quotient[r+1, 2]]][Mod[k+1, 2]]];
Table[T[n, k], {n, 0, 7}, {k, 1, Max[1, 2^(n-1)]}] // Flatten (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)
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PROG
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(Haskell)
data Dtree = Dtree Dtree (Integer, Integer) Dtree
a230871 n k = a230871_tabf !! n !! k
a230871_row n = a230871_tabf !! n
a230871_tabf = [0] : map (map snd) (rows $ deleham (0, 1)) where
rows (Dtree left (x, y) right) =
[(x, y)] : zipWith (++) (rows left) (rows right)
deleham (x, y) = Dtree
(deleham (y, y + x)) (x, y) (deleham (y, 3 * y - x))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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