A346504 -id:A346504 - 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).

A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 7, 14, 26, 43, 79, 148, 264, 483, 903, 1664, 3080, 5771, 10795, 20209, 38059, 71799, 135569, 256762, 487310, 925981, 1762841, 3361897, 6419595, 12275301, 23505143, 45061424, 86485016, 166176499, 319630115, 615387675, 1185940209, 2287527119, 4416083429

Offset: 0

Ilya Gutkovskiy, Jul 21 2021

nonn

Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.

nmax = 40; A[] = 0; Do[A[x] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[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, 40}]

a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 30 2021
From Seiichi Manyama, Sep 26 2024: (Start)
G.f.: 2/(1 + sqrt(1 - 4*x^3/(1 - x))).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)

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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A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

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Author

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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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