A376718 -id:A376718 - cp's OEIS frontend

A376716 Expansion of (1 - x + x^2)/((1 - x + x^2)^2 - 4*x^2).

1, 1, 4, 11, 27, 72, 189, 493, 1292, 3383, 8855, 23184, 60697, 158905, 416020, 1089155, 2851443, 7465176, 19544085, 51167077, 133957148, 350704367, 918155951, 2403763488, 6293134513, 16475640049, 43133785636, 112925716859, 295643364939, 774004377960

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

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

Cf. A000302, A376717, A376718.

Cf. A182890.

my(N=30, x='x+O('x^N)); Vec((1-x+x^2)/((1-x+x^2)^2-4*x^2))

a(n) = sum(k=0, n\2, binomial(2*n-2*k+1, 2*k));

a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-2*k+1,2*k).

A376717 Expansion of (1 - x + x^3)/((1 - x + x^3)^2 - 4*x^3).

Original entry on oeis.org

1, 1, 1, 4, 11, 22, 42, 91, 205, 443, 936, 1999, 4316, 9300, 19949, 42785, 91917, 197548, 424331, 911218, 1957086, 4203927, 9029949, 19395031, 41657808, 89477119, 192189304, 412803240, 886657081, 1904448737, 4090567673, 8786130132, 18871714923, 40534529294

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,2,0,-1).

Cf. A000302, A376716, A376718.

Cf. A375278, A375279.

my(N=40, x='x+O('x^N)); Vec((1-x+x^3)/((1-x+x^3)^2-4*x^3))

a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, 2*k));

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,2*k).

A376786 Expansion of (1 + x - x^4)/((1 + x - x^4)^2 - 4*x).

Original entry on oeis.org

1, 3, 5, 7, 10, 21, 48, 99, 183, 326, 602, 1165, 2282, 4396, 8318, 15675, 29743, 56841, 108765, 207510, 394809, 750880, 1429845, 2725685, 5196420, 9901692, 18859649, 35921156, 68432064, 130388316, 248437405, 473322419, 901717453, 1717851555, 3272777450

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,2,0,0,-1).

Cf. A099511, A376785.

Cf. A375282, A375283, A376718.

my(N=40, x='x+O('x^N)); Vec((1+x-x^4)/((1+x-x^4)^2-4*x))

a(n) = sum(k=0, n\4, binomial(2*n-6*k+1, 2*k+1));

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) + 2*a(n-5) - a(n-8).

a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-6*k+1,2*k+1).

cp's OEIS Frontend

A376716 Expansion of (1 - x + x^2)/((1 - x + x^2)^2 - 4*x^2).

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A376717 Expansion of (1 - x + x^3)/((1 - x + x^3)^2 - 4*x^3).

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A376786 Expansion of (1 + x - x^4)/((1 + x - x^4)^2 - 4*x).

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