A346503 -id:A346503 - cp's OEIS frontend

A357308 a(0) = a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

0, 0, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 24, 39, 67, 116, 196, 324, 534, 892, 1516, 2601, 4463, 7630, 13022, 22276, 38286, 66084, 114328, 197929, 342783, 594218, 1031794, 1794944, 3127450, 5455272, 9523812, 16640542, 29102938, 50951070, 89289998, 156616648, 274923328, 482945930, 848972814

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Cf. A006318, A023426, A023431, A025246, A346503, A346504, A357307.

a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = x^2 * (1 + x * A(x)^2) / (1 - x).

A346504 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 3, 4, 6, 14, 28, 49, 95, 196, 386, 754, 1524, 3102, 6258, 12700, 26032, 53440, 109772, 226457, 468863, 972300, 2020274, 4208530, 8784556, 18365322, 38461110, 80682740, 169501696, 356579216, 751138916, 1584281062, 3345404514, 7072055268, 14965933024, 31702754496

Offset: 0

Ilya Gutkovskiy, Jul 21 2021

nonn

Cf. A023426, A023431, A025243, A090345, A248100, A257290, A346503.

nmax = 37; A[] = 0; Do[A[x] = 1 + x + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 37}]
CoefficientList[Series[(1 - x)*(1 - Sqrt[(1 - x - 4*x^3 - 4*x^4)/(1 - x)]) / (2*x^3), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

a(0) = a(1) = 1, a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

G.f.: (1-x)*(1 - sqrt((1 - x - 4*x^3 - 4*x^4)/(1-x))) / (2*x^3). - Vaclav Kotesovec, Sep 27 2023

A357307 a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 8, 13, 25, 49, 91, 177, 349, 681, 1349, 2693, 5377, 10806, 21820, 44163, 89721, 182868, 373616, 765341, 1571551, 3233690, 6667242, 13772469, 28498419, 59065838, 122606998, 254865837, 530507839, 1105663034, 2307131590, 4819623077, 10079039819, 21099213611, 44211213545

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Cf. A000245, A023426, A023431, A090345, A346503, A346504, A357308.

a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 38}]
nmax = 38; A[] = 0; Do[A[x] = 1 + x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + x^2 * (1 + x * A(x)^2) / (1 - x).

cp's OEIS Frontend

A357308 a(0) = a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

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A346504 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x)^2 / (1 - x).

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Crossrefs

Programs

Mathematica

Formula

A357307 a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

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