Displaying 51-60 of 208 results found.
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Powers of 48: a(n) = 48^n.
+10
11
1, 48, 2304, 110592, 5308416, 254803968, 12230590464, 587068342272, 28179280429056, 1352605460594688, 64925062108545024, 3116402981210161152, 149587343098087735296, 7180192468708211294208, 344649238497994142121984, 16543163447903718821855232
COMMENTS
Same as Pisot sequences E(1, 48), L(1, 48), P(1, 48), T(1, 48). Essentially same as Pisot sequences E(48, 2304), L(48, 2304), P(48, 2304), T(48, 2304). See A008776 for definitions of Pisot sequences. If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4,5,6,7} such that for fixed y_1,y_2,...,y_n in {1,2,3,4,5,6,7} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007 The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 48-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
PROG
(Python) for n in range(0, 20): print(48**n, end=', ') # Stefano Spezia, Nov 21 2018
Pisot sequences E(4,8), L(4,8), P(4,8), T(4,8).
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4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
FORMULA
a(n) = 2^(n+2).
a(n) = 2*a(n-1).
CROSSREFS
Subsequence of A000079. See A008776 for definitions of Pisot sequences.
T(n,k)=Number of nXk 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally, diagonally or antidiagonally
+10
11
3, 6, 9, 12, 18, 27, 24, 54, 54, 81, 48, 144, 246, 162, 243, 96, 396, 912, 1122, 486, 729, 192, 1080, 3612, 5808, 5118, 1458, 2187, 384, 2952, 13992, 33702, 37008, 23346, 4374, 6561, 768, 8064, 54600, 186720, 316800, 235824, 106494, 13122, 19683, 1536, 22032
COMMENTS
Table starts
.....3......6.......12........24..........48............96............192
.....9.....18.......54.......144.........396..........1080...........2952
....27.....54......246.......912........3612.........13992..........54600
....81....162.....1122......5808.......33702........186720........1054446
...243....486.....5118.....37008......316800.......2515716.......20706696
...729...1458....23346....235824.....2986152......33994188......409408542
..2187...4374...106494...1502736....28178262.....459797904.....8119777890
..6561..13122...485778...9575856...266016264....6221092260...161274860934
.19683..39366..2215902..61020048..2511769872...84180552504..3205524631536
.59049.118098.10107954.388836912.23718269934.1139126465856.63736076920680
FORMULA
Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 3*a(n-1)
k=3: a(n) = 5*a(n-1) -2*a(n-2)
k=4: a(n) = 7*a(n-1) -4*a(n-2)
k=5: a(n) = 14*a(n-1) -45*a(n-2) +15*a(n-3) +36*a(n-4) -19*a(n-5) +2*a(n-6)
k=6: [order 9]
k=7: [order 19]
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 2*a(n-1) +2*a(n-2)
n=3: a(n) = 3*a(n-1) +4*a(n-2) -2*a(n-3) for n>4
n=4: a(n) = 3*a(n-1) +16*a(n-2) -3*a(n-3) -25*a(n-4) +2*a(n-5) +4*a(n-6) for n>7
n=5: [order 10] for n>11
n=6: [order 23] for n>24
n=7: [order 46] for n>47
EXAMPLE
Some solutions for n=4 k=4
..0..0..1..0....0..0..1..0....0..0..1..0....0..0..1..0....1..0..0..1
..0..0..1..2....0..0..1..2....1..0..1..0....1..0..1..0....1..0..0..1
..1..0..1..2....0..0..1..0....1..0..1..2....0..0..1..2....0..0..0..0
..0..0..1..2....0..0..0..0....0..0..1..0....1..0..1..2....1..0..1..0
1, 43, 1849, 79507, 3418801, 147008443, 6321363049, 271818611107, 11688200277601, 502592611936843, 21611482313284249, 929293739471222707, 39959630797262576401, 1718264124282290785243, 73885357344138503765449
COMMENTS
Same as Pisot sequences E(1, 43), L(1, 43), P(1, 43), T(1, 43). Essentially same as Pisot sequences E(43, 1849), L(43, 1849), P(43, 1849), T(43, 1849). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 43-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011 Numbers n such that sigma(43*n) = 43*n + sigma(n). - Jahangeer Kholdi, Nov 24 2013
1, 47, 2209, 103823, 4879681, 229345007, 10779215329, 506623120463, 23811286661761, 1119130473102767, 52599132235830049, 2472159215084012303, 116191483108948578241, 5460999706120583177327
COMMENTS
Same as Pisot sequences E(1, 47), L(1, 47), P(1, 47), T(1, 47). Essentially same as Pisot sequences E(47, 2209), L(47, 2209), P(47, 2209), T(47, 2209). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 47-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011 Numbers n such that sigma(47*n) = 47*n + sigma(n). - Jahangeer Kholdi, Nov 24 2013
Pisot sequences E(7,9), P(7,9).
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7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081, 922111, 1221537
FORMULA
a(n) = a(n-2) + a(n-3) for n>=3. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016
G.f.: (7+9*x+5*x^2) / (1-x^2-x^3). - Colin Barker, Jun 05 2016
MATHEMATICA
CoefficientList[Series[(7 + 9 x + 5 x^2)/(1 - x^2 - x^3), {x, 0, 50}], x] (* Stefano Spezia, Aug 31 2018 *)
CROSSREFS
See A008776 for definitions of Pisot sequences.
5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593
COMMENTS
An Engel expansion of 1/2 to the base 2 as defined in A181565, with the associated series expansion 1/2 = 2/5 + 2^2/(5*9) + 2^3/(5*9*17) + 2^4/(5*9*17*33) + ... . - Peter Bala, Oct 28 2013
FORMULA
a(n) = 2^(n+2) + 1.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: -(6*x-5) / ((x-1)*(2*x-1)). - Colin Barker, Jun 21 2014
MATHEMATICA
LinearRecurrence[{3, -2}, {5, 9}, 40] (* Harvey P. Dale, Jun 10 2015 *)
PROG
(PARI) Vec(-(6*x-5)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 21 2014
CROSSREFS
Subsequence of A000051. See A008776 for definitions of Pisot sequences.
Numbers n such that A081249(m)/m^2 has a local maximum for m = n.
+10
10
2, 6, 20, 60, 182, 546, 1640, 4920, 14762, 44286, 132860, 398580, 1195742, 3587226, 10761680, 32285040, 96855122, 290565366, 871696100, 2615088300, 7845264902, 23535794706, 70607384120, 211822152360, 635466457082, 1906399371246
COMMENTS
The limit of the local maxima, lim A081249(n)/n^2 = 1/6. For local minima cf. A081250. Also the number of different 4- and 3-colorings for the vertices of all triangulated planar polygons on a base with n+2 vertices, if the colors of the two base vertices are fixed. - Patrick Labarque, Mar 23 2010 a(n) = the number of ternary sequences of length n+1 where the numbers of (0's, 1's) are both odd.
A015518 covers the (odd, even) and (even, odd) cases, and A122983 covers (even, even). (End)
FORMULA
G.f.: 2/((1-x)*(1+x)*(1-3*x)).
a(n) = a(n-2) + 2*3^(n) for n > 1.
a(n) = (9*3^(n-1) -(-1)^n -2)/4.
E.g.f.: (3*exp(3*x) - 2*exp(x) - exp(-x))/4. (End)
EXAMPLE
6 is a term since A081249(5)/5^2 = 4/25 = 0.160, A081249(6)/6^2 = 7/36 = 0.194, A081249(7)/7^2 = 9/49 = 0.184.
MATHEMATICA
a[n_]:= Floor[3^(n+1)/4]; Array[a, 30]
Table[(9*3^(n-1) -(-1)^n -2)/4, {n, 1, 30}] (* G. C. Greubel, Jul 14 2019 *)
PROG
(PARI) vector(30, n, (9*3^(n-1) -(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019 (Sage) [(9*3^(n-1) -(-1)^n -2)/4 for n in (1..30)] # G. C. Greubel, Jul 14 2019 (GAP) List([1..30], n-> (9*3^(n-1) -(-1)^n -2)/4) # G. C. Greubel, Jul 14 2019
4, 5, 7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143
FORMULA
a(n) = Fib(n+3)+2 = A020743(n-2) = A157725(n+3); a(n) = 2a(n-1) - a(n-3). G.f.: -(-4+3*x+3*x^2)/(x-1)/(x^2+x-1) = -2/(x-1)+(-x-2)/(x^2+x-1) . - R. J. Mathar, Nov 23 2007
PROG
(PARI) pisotL(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
a
}
CROSSREFS
See A008776 for definitions of Pisot sequences.
2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141
FORMULA
a(n) = Fibonacci(n+3); a(n) = a(n-1) + a(n-2).
a(n) = (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5))+(1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016 E.g.f.: 2*(2*sqrt(5)*sinh(sqrt(5)*x/2) + 5*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, Jun 05 2016
MATHEMATICA
LinearRecurrence[{1, 1}, {2, 3}, 40] (* or *) Fibonacci[Range[3, 50]] (* Harvey P. Dale, Nov 22 2012 *)
PROG
(PARI) Vec((2+x)/(1-x-x^2) + O(x^40)) \\ Colin Barker, Jun 05 2016 (GAP)
CROSSREFS
Subsequence of A000045. See A008776 for definitions of Pisot sequences. See A000045 for the Fibonacci numbers.
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