A024494 -id:A024494 - cp's OEIS frontend

A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

1, 1, 3, 1, 2, 7, 1, 2, 4, 15, 1, 2, 3, 8, 31, 1, 2, 3, 5, 16, 63, 1, 2, 3, 4, 10, 32, 127, 1, 2, 3, 4, 6, 21, 64, 255, 1, 2, 3, 4, 5, 12, 43, 128, 511, 1, 2, 3, 4, 5, 7, 28, 86, 256, 1023, 1, 2, 3, 4, 5, 6, 14, 64, 171, 512, 2047, 1, 2, 3, 4, 5, 6, 8, 36, 136, 341, 1024, 4095

Offset: 0

Seiichi Manyama, Mar 22 2019

			Square array begins:
     1,   1,   1,   1,  1,  1,  1,  1, 1, ...
     3,   2,   2,   2,  2,  2,  2,  2, 2, ...
     7,   4,   3,   3,  3,  3,  3,  3, 3, ...
    15,   8,   5,   4,  4,  4,  4,  4, 4, ...
    31,  16,  10,   6,  5,  5,  5,  5, 5, ...
    63,  32,  21,  12,  7,  6,  6,  6, 6, ...
   127,  64,  43,  28, 14,  8,  7,  7, 7, ...
   255, 128,  86,  64, 36, 16,  9,  8, 8, ...
   511, 256, 171, 136, 93, 45, 18, 10, 9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.

Cf. A306846, A306915, A307079.

T[n_, k_] := Sum[Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+1,k*j+1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i,k*j) * binomial(n-i,k*j).

A138635 a(n) =3a(n-3)-3a(n-6)+2*a(n-9).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 2, 6, 5, 5, 11, 10, 11, 21, 21, 22, 42, 43, 43, 85, 86, 85, 171, 171, 170, 342, 341, 341, 683, 682, 683, 1365, 1365, 1366, 2730, 2731, 2731, 5461, 5462, 5461, 10923, 10923, 10922, 21846, 21845, 21845, 43691, 43690, 43691, 87381

Offset: 0

Paul Curtz, May 14 2008

As the recurrence shows, these are three interleaved sequences which obey recurrences b(n)=3*b(n-1)-3*b(n-2)+2*b(n-3), indicating that the b(n) equal their third differences.
These three sequences are A024495, A024494 (or A131708) and A024493 (or A130781).
Their starting "vectors" b(0,1,2) are 0,0,1 and 0,1,2 and 1,1,1, respectively, therefore linearly independent, such that other sequences with the same recursion as b(n) can be written as linear combinations of these.

Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,2).

Cf. A024493, A024494, A024495, A131708, A130781.

a(18*n) = 21*A133853(n).

G.f.: -x^2*(1+x^2-2*x^3+x^4-x^5+x^6)/((2*x^3-1)*(x^6-x^3+1)). - R. J. Mathar, May 17 2009

Edited by R. J. Mathar, May 17 2009

A139468 a(n) = Sum{k=0..n} C(n,3k+1)^2.

Original entry on oeis.org

0, 1, 4, 9, 17, 50, 261, 1275, 5028, 17253, 58601, 218042, 876789, 3537847, 13783018, 52301709, 198627921, 767778786, 3010327497, 11824753551, 46200429186, 179787741723, 700285942731, 2738134757118, 10739885115573, 42164261091351, 165467386466802

Offset: 0

N. J. A. Sloane, Jun 12 2008

nonn

The recurrence is same as for A119363. - Vaclav Kotesovec, Mar 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A024494, A119363, A139469.

Table[Sum[Binomial[n,3k+1]^2,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Sep 08 2018 *)

a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 12 2019

A102517 Expansion of (1+x^2)/((1-x+x^2)(1+2x^2)).

Original entry on oeis.org

1, 1, -1, -2, 1, 3, -2, -5, 5, 10, -11, -21, 22, 43, -43, -86, 85, 171, -170, -341, 341, 682, -683, -1365, 1366, 2731, -2731, -5462, 5461, 10923, -10922, -21845, 21845, 43690, -43691, -87381, 87382, 174763, -174763, -349526, 349525, 699051, -699050, -1398101, 1398101, 2796202, -2796203, -5592405

Offset: 0

Paul Barry, Jan 13 2005

Index entries for linear recurrences with constant coefficients, signature (1,-3,2,-2).

Cf. A001045, A078008.

CoefficientList[Series[(1+x^2)/((1-x+x^2)(1+2x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,-3,2,-2},{1,1,-1,-2},50] (* Harvey P. Dale, Oct 28 2011 *)

G.f.: (1+x^2)^2/((1+x^2)^3+x^6)+x(1+x^2)/((1+x^2)^3+x^6).
a(n) = Sum_{k=0..floor(n/2)} T(n-k, k)*(-1)^k, T(n, k) = Sum_{i=0..k} C(n, i) (A008949).
a(n) = (-1)^(n/2)*(Sum_{k=0..floor(n/6)} C(n/2, 3*k))*(1+(-1)^n)/2 + (-1)^((n-1)/2)*(Sum_{k=0..floor((n+1)/6)} C((n+1)/2, 3*k+1))*(1-(-1)^n)/2.
a(n) = 2^(n/2)*(cos(Pi*n/2)/3+sqrt(2)*sin(Pi*n/2)/3)+cos(Pi*n/3+Pi/3)/3+sqrt(3)*sin(Pi*n/3+Pi/3)/3.
a(2*n) = (-1)^n*A024493(n); a(2*n+1) = (-1)^n*A024494(n).
a(0)=1, a(1)=1, a(2)=-1, a(3)=-2, a(n) = a(n-1)-3*a(n-2)+2*a(n-3)-2*a(n-4). - Harvey P. Dale, Oct 28 2011

A080850 Number triangle related to a problem of Knuth.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 5, 6, 4, 1, 11, 10, 11, 10, 5, 1, 22, 21, 21, 21, 15, 6, 1, 43, 43, 42, 42, 36, 21, 7, 1, 85, 86, 85, 84, 78, 57, 28, 8, 1, 170, 171, 171, 169, 162, 135, 85, 36, 9, 1, 341, 341, 342, 340, 331, 297, 220, 121, 45, 10, 1, 683, 682, 683, 671, 628, 517

Offset: 1

Paul Barry, Feb 20 2003

In lower-triangular form, the columns are the binomial transforms of the sequences with g.f. x^(k-1)/(1-x^3). The first three columns are A024493, A024494, A024495.

			Rows are {1}, {1,1}, {1,2,1}, {2,3,3,1}, {5,5,6,4,1}, {11,10,11,10,5,1}...

T(n, 1) = A024493(n). T(n, k)=0, k>n, T(n, n)=1. T(n, k) = T(n-1, k-1)+T(n-1, k).

A375169 Expansion of (1 - x) / ((1 - x)^3 - x^4).

Original entry on oeis.org

1, 2, 3, 4, 6, 11, 22, 43, 80, 144, 257, 462, 839, 1532, 2798, 5099, 9274, 16855, 30640, 55728, 101393, 184490, 335659, 610628, 1110790, 2020635, 3675822, 6686979, 12164896, 22130208, 40258737, 73237462, 133231279, 242370396, 440913550, 802098203, 1459155634

Offset: 0

Seiichi Manyama, Aug 05 2024

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1).

Cf. A024494, A052921, A124820, A137357.

Cf. A003522, A107068.

my(N=40, x='x+O('x^N)); Vec((1-x)/((1-x)^3-x^4))

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+1-k,n-4*k).
a(n) = (n + 1)*hypergeom([(1-n)/4, (2-n)/4, (3-n)/4, -n/4], [2/3, 4/3, -1-n], -4^4/3^3). - Stefano Spezia, Jun 18 2025

cp's OEIS Frontend

A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

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A138635 a(n) =3*a(n-3)-3*a(n-6)+2*a(n-9).

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A139468 a(n) = Sum{k=0..n} C(n,3k+1)^2.

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A102517 Expansion of (1+x^2)/((1-x+x^2)*(1+2*x^2)).

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Mathematica

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A080850 Number triangle related to a problem of Knuth.

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A375169 Expansion of (1 - x) / ((1 - x)^3 - x^4).

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Formula

A138635 a(n) =3a(n-3)-3a(n-6)+2*a(n-9).

A102517 Expansion of (1+x^2)/((1-x+x^2)(1+2x^2)).