Search: a081341 -id:a081341
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1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256
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OFFSET
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0,4
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COMMENTS
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Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence. For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - Geoffrey Critzer, Apr 06 2013
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LINKS
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FORMULA
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a(n,k) = 2^(n-1)*C(n, k), for n>0.
G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Feb 05 2012
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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 4, 2;
4, 12, 12, 4;
8, 32, 48, 32, 8;
...
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MATHEMATICA
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nn=10; f[list_]:=Select[list, #>0&]; a=(x+y x)/(1-(x+y x)); Map[f, CoefficientList[Series[1/(1-a), {x, 0, nn}], {x, y}]]//Grid (* Geoffrey Critzer, Apr 06 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A134309
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Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).
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+10
14
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1, 0, 1, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,6
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COMMENTS
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As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).
Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007
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LINKS
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FORMULA
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Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Feb 04 2012
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EXAMPLE
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Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
0, 1;
0, 0, 2;
0, 0, 0, 4;
0, 0, 0, 0, 8;
0, 0, 0, 0, 0, 16;
...
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MATHEMATICA
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Join[{1}, Flatten[Table[Join[{PadRight[{}, n], 2^(n-1)}], {n, 20}]]] (* Harvey P. Dale, Jan 04 2024 *)
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PROG
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CROSSREFS
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Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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1, 4, 25, 172, 1201, 8404, 58825, 411772, 2882401, 20176804, 141237625, 988663372, 6920643601, 48444505204, 339111536425, 2373780754972, 16616465284801, 116315256993604, 814206798955225, 5699447592686572
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OFFSET
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0,2
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COMMENTS
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Number of compositions of even natural numbers into n parts <=6. [Adi Dani, May 28 2011]
a(n)+(a(n)+1)+...+(a(n+1)-7^n-1)=(a(n+1)-7^n)+...+(a(n+1)-1). Let
S(2n) and S(2n+1) be the sets of addends on the left- and right-hand
sides, respectively, of the preceding equations. Then, since the
intersection of any 2 different S(i) is null and the union of all of
them is the positive integers, {S(i)} forms a partition of the
positive integers. See also A034659.
In general, for k>0, let b(n)=((4k+3)^n+1)/2. Then b(n)+(b(n)+1)+ ...
+(b(n+1)-(4k+3)^n-1)=k*((b(n+1)-(4k+3)^n)+ ... +(b(n+1)-1)). Then,
for each k, the set of addends on the two sides of these equations
also forms a partition of the positive integers. Also, with b(0)=1,
b(n)=(4k+3)*b(n-1)-(2k+1).
For k>0, let c(0)=1 and, for n>0, c(n)=(2*(2k+1))^n/2. Then the
sequence b(0),b(1),... is the binomial transform of the sequence
c(0),c(1),....
For k>0, let d(2n)=(2k+1)^(2n) and d(2n+1)=0. Then the sequence
b(0),b(1),... is the (2k+2)nd binomial transform of the sequence
d(0),d(1),.... (End)
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LINKS
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FORMULA
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E.g.f.: exp(4*x)*cosh(3*x). - Paul Barry, Apr 20 2003
a(n) = 7a(n-1) - 3, a(0) = 1.
a(n) = ((4+sqrt(9))^n+(4-sqrt(9))^n)/2. [Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008]
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EXAMPLE
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a(2)=25: there are 25 compositions of even numbers into 2 parts <=6:
(0,0)
(0,2),(2,0),(1,1)
(0,4),(4,0),(1,3),(3,1),(2,2)
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3)
(2,6),(6,2),(3,5),(5,3),(4,4)
(4,6),(6,4),(5,5)
(6,6)
(end)
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MAPLE
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PROG
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(PARI) Vec((1-4*x)/((1-x)*(1-7*x)) + O(x^100)) \\ Altug Alkan, Nov 01 2015
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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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A166469
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Number of divisors of n which are not multiples of consecutive primes.
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+10
12
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1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 3, 5, 2, 4, 2, 6, 4, 4, 2, 5, 3, 4, 4, 6, 2, 5, 2, 6, 4, 4, 3, 5, 2, 4, 4, 8, 2, 6, 2, 6, 4, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 8, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 6, 2, 6, 2, 4, 4, 6, 3, 6, 2, 10, 5, 4, 2, 8, 4, 4, 4, 8, 2, 6, 4, 6, 4, 4, 4, 7, 2, 6, 6, 9, 2, 6, 2, 8, 5
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OFFSET
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1,2
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COMMENTS
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Links various subsequences of A025487 with an unusual number of important sequences, including the Fibonacci, Lucas, and other generalized Fibonacci sequences (see cross-references).
If a number is a product of any number of consecutive primes, the number of its divisors which are not multiples of n consecutive primes is always a Fibonacci n-step number. See also A073485, A167447.
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LINKS
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FORMULA
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a) If n has no prime gaps in its factorization (cf. A073491), then, if the canonical factorization of n into prime powers is the product of p_i^(e_i), a(n) is the sum of all products of one or more nonadjacent exponents, plus 1. For example, if A001221(n) = 3, a(n) = e_1*e_3 + e_1 + e_2 + e_3 + 1. If A001221(n) = k, the total number of terms always equals A000045(k+2).
The answer can also be computed in k steps, by finding the answers for the products of the first i powers, for i = 1 to i = k. Let the result of the i-th step be called r(i). r(1) = e_1 + 1; r(2) = e_1 + e_2 +1; for i > 2, r(i) = r(i-1) + e_i * r(i-2).
b) If n has prime gaps in its factorization, express it as a product of the minimum number of A073491's members possible. Then apply either of the above methods to each of those members, and multiply the results to get a(n). a(n) = A000005(n) iff n has no pair of consecutive primes as divisors.
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EXAMPLE
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Since 3 of 30's 8 divisors (6, 15, and 30) are multiples of 2 or more consecutive primes, a(30) = 8 - 3 = 5.
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MATHEMATICA
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Array[DivisorSum[#, 1 &, FreeQ[Differences@ PrimePi@ FactorInteger[#][[All, 1]], 1] &] &, 105] (* Michael De Vlieger, Dec 16 2017 *)
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PROG
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(PARI)
A296210(n) = { if(1==n, return(0)); my(ps=factor(n)[, 1], pis=vector(length(ps), i, primepi(ps[i])), diffsminusones = vector(length(pis)-1, i, (pis[i+1]-pis[i])-1)); !factorback(diffsminusones); };
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488
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OFFSET
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0,1
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COMMENTS
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Essentially first differences of A125682.
Binomial transform of A005053 without initial term 1.
Second binomial transform of A164346.
Inverse binomial transform of A169634.
Second inverse binomial transform of A103333 without initial term 1.
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LINKS
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FORMULA
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a(n) = 6*a(n-1) for n > 0; a(0) = 3.
G.f.: 3/(1-6*x).
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PROG
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(Magma) [ 3*6^n: n in [0..19] ];
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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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1, 2, 13, 62, 313, 1562, 7813, 39062, 195313, 976562, 4882813, 24414062, 122070313, 610351562, 3051757813, 15258789062, 76293945313, 381469726562, 1907348632813, 9536743164062, 47683715820313, 238418579101562
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A003665. 2nd binomial transform of (1,0,9,0,81,0,729,0,..). Case k=2 of family of recurrences a(n)=2k*a(n-1)-(k^2-9)*a(n-2), a(0)=0, a(1)=k. A003665 is case k=1.
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LINKS
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FORMULA
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a(n) = 4*a(n-1) + 5*a(n-2), a(0)=1, a(1)=2.
G.f.: (1-2*x)/((1+x)*(1-5*x)).
E.g.f.: exp(2*x) * cosh(3*x).
a(n) = ((2+sqrt(9))^n+(2-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
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MATHEMATICA
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CoefficientList[Series[(1 - 2 x) / ((1 + x) (1 - 5 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)
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PROG
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(Sage) [lucas_number2(n, 4, -5)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009
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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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A100852
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Triangle read by rows: T(n,k) = 2^k * 3^n, 0 <= k <= n.
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+10
3
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1, 3, 6, 9, 18, 36, 27, 54, 108, 216, 81, 162, 324, 648, 1296, 243, 486, 972, 1944, 3888, 7776, 729, 1458, 2916, 5832, 11664, 23328, 46656, 2187, 4374, 8748, 17496, 34992, 69984, 139968, 279936, 6561, 13122, 26244, 52488, 104976, 209952, 419904, 839808
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1;
3, 6;
9, 18, 36;
27, 54, 108, 216;
81, 162, 324, 648, 1296;
...
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MATHEMATICA
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PROG
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(PARI) for(n=0, 8, for(k=0, n, print1(2^k*3^n", "))) \\ Satish Bysany, Mar 06 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191347
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Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.
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+10
3
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1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1
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OFFSET
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0,8
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LINKS
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FORMULA
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For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019
T(n, k) = Sum_(i=0, floor((k+1)/2), binomial(k, 2*i)*floor(sqrtint(n))^(k-2*i)*n^i)) for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019
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EXAMPLE
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1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, ...
1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240, 11584, ...
1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...
1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, ...
1, 2, 10, 44, 196, 872, 3880, 17264, 76816, 341792, 1520800, ...
1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, ...
1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, ...
1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...
1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, ...
1, 3, 20, 126, 796, 5028, 31760, 200616, 1267216, 8004528, 50561600, ...
1, 3, 21, 135, 873, 5643, 36477, 235791, 1524177, 9852435, 63687141, ...
1, 3, 22, 144, 952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...
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PROG
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(PARI) T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
(PARI) T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n, k-2)*(n-T(n, 1)^2) + T(n, k-1)*T(n, 1)*2));
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CROSSREFS
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Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Cf. A191348 which uses ceiling() in place of floor().
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191348
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Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.
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+10
3
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1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019
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EXAMPLE
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1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, ...
1, 2, 7, 26, 97, 362, 1351, 5042, 18817, 70226, 262087, ...
1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...
1, 3, 14, 72, 376, 1968, 10304, 53952, 282496, 1479168, 7745024, ...
1, 3, 15, 81, 441, 2403, 13095, 71361, 388881, 2119203, 11548575, ...
1, 3, 16, 90, 508, 2868, 16192, 91416, 516112, 2913840, 16450816, ...
1, 3, 17, 99, 577, 3363, 19601, 114243, 665857, 3880899, 22619537, ...
1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...
1, 4, 26, 184, 1316, 9424, 67496, 483424, 3462416, 24798784, 177615776, ...
1, 4, 27, 196, 1433, 10484, 76707, 561236, 4106353, 30044644, 219825387, ...
1, 4, 28, 208, 1552, 11584, 86464, 645376, 4817152, 35955712, 268377088, ...
1, 4, 29, 220, 1673, 12724, 96773, 736012, 5597777, 42574180, 323800109, ...
1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...
...
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PROG
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(PARI) T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n, k-2)*(n-T(n, 1)^2) + T(n, k-1)*T(n, 1)*2));
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CROSSREFS
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Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.
Cf. A191347 which uses floor() in place of ceiling().
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KEYWORD
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AUTHOR
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STATUS
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approved
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A192396
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Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.
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+10
2
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0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 4, 4, 2, 0, 0, 8, 13, 8, 2, 0, 0, 16, 40, 32, 12, 3, 0, 0, 32, 121, 128, 62, 18, 3, 0, 0, 64, 364, 512, 312, 108, 24, 4, 0, 0, 128, 1093, 2048, 1562, 648, 171, 32, 4, 0, 0, 256, 3280, 8192, 7812, 3888, 1200, 256, 40, 5, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,8
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COMMENTS
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T(n,k) is the number of compositions of odd natural numbers into n parts <=k.
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LINKS
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EXAMPLE
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T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4
1: (0,1), (1,0);
3: (1,2), (2,1), (0,3), (3,0);
5: (1,4), (4,1), (2,3), (3,2);
7: (3,4), (4,3).
The table starts
0, 4, 13, 32, 62, 108, ... A036487;
0, 8, 40, 128, 312, 648, ... A191903;
0, 16, 121, 512, 1562, 3888, ... A191902;
. . . . ...
Antidiagonal triangle begins:
0;
0, 0;
0, 1, 0;
0, 2, 1, 0;
0, 4, 4, 2, 0;
0, 8, 13, 8, 2, 0;
0, 16, 40, 32, 12, 3, 0;
0, 32, 121, 128, 62, 18, 3, 0;
0, 64, 364, 512, 312, 108, 24, 4, 0;
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MAPLE
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A192396 := proc(n, k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:
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MATHEMATICA
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T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];
Table[T[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten
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PROG
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(Magma)
A192396:= func< n, k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;
(SageMath)
def A192396(n, k): return ((k+1)^n - ((k+1)%2))//2
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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