A346049 -id:A346049 - cp's OEIS frontend

A346047 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=1..n-3} a(k) * a(n-k-3).

1, 1, 1, 0, 1, 2, 2, 2, 4, 8, 10, 13, 24, 42, 61, 90, 156, 265, 410, 646, 1093, 1834, 2948, 4789, 8050, 13475, 22129, 36570, 61435, 103039, 171384, 286156, 481691, 810502, 1359194, 2284789, 3856974, 6512001, 10982193, 18550116, 31406597, 53194727, 90082902

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

Cf. A025250, A050253, A307970, A343304, A346048, A346049.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[k] a[n - k - 3], {k, 1, n - 3}]; Table[a[n], {n, 0, 42}]
nmax = 42; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

a(n) ~ sqrt((3 - 3*r^3 - 4*r^4 - 2*r^5)/(8*Pi)) / (n^(3/2) * r^(n+3)), where r = 0.5701490701528437821032230160646366013461622472504286581627... is the root of the equation 1 - 2*r^3 - 4*r^4 - 4*r^5 + r^6 = 0. - Vaclav Kotesovec, Jul 03 2021

A346048 a(0) = ... = a(3) = 1; a(n) = Sum_{k=1..n-4} a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 3, 3, 3, 5, 9, 15, 19, 24, 35, 59, 95, 137, 191, 280, 445, 706, 1071, 1575, 2357, 3663, 5755, 8890, 13483, 20518, 31759, 49658, 77267, 119135, 183523, 284793, 444883, 694798, 1080865, 1679142, 2616399, 4092497, 6408249, 10021176, 15657643

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

Cf. A025250, A307971, A343304, A343305, A346047, A346049.

a:= proc(n) option remember; `if`(n<4, 1,
      add(a(j)*a(n-4-j), j=1..n-4))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 03 2021

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[a[k] a[n - k - 4], {k, 1, n - 4}]; Table[a[n], {n, 0, 45}]
nmax = 45; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x) * (A(x) - 1).

A346077 a(n) = 1 + Sum_{k=1..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 12, 18, 26, 36, 49, 69, 101, 150, 221, 320, 460, 667, 981, 1456, 2161, 3191, 4698, 6932, 10283, 15324, 22870, 34103, 50813, 75770, 113229, 169590, 254340, 381579, 572537, 859511, 1291681, 1943489, 2926980, 4410709, 6649220, 10028570

Offset: 0

Ilya Gutkovskiy, Jul 04 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A105633, A217282, A307972, A346049, A346074, A346075, A346076.

a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 1, n - 5}]; Table[a[n], {n, 0, 47}]
nmax = 47; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

@CachedFunction
def a(n): # a = A346077
    if (n<6): return 1
    else: return 1 + sum(a(k)*a(n-k-5) for k in range(1,n-4))
[a(n) for n in range(51)] # G. C. Greubel, Nov 27 2022

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x) * (A(x) - 1).

cp's OEIS Frontend

A346047 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=1..n-3} a(k) * a(n-k-3).

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A346048 a(0) = ... = a(3) = 1; a(n) = Sum_{k=1..n-4} a(k) * a(n-k-4).

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A346077 a(n) = 1 + Sum_{k=1..n-5} a(k) * a(n-k-5).

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