A050253 -id:A050253 - cp's OEIS frontend

A343304 a(0) = a(1) = a(2) = 1; a(n) = a(n-3) + Sum_{k=0..n-4} a(k) * a(n-k-4).

1, 1, 1, 1, 2, 3, 4, 6, 10, 16, 25, 40, 66, 109, 179, 296, 495, 831, 1396, 2353, 3985, 6770, 11523, 19657, 33621, 57633, 98969, 170245, 293371, 506371, 875284, 1515029, 2625842, 4556806, 7916943, 13769900, 23975073, 41785251, 72894759, 127279673, 222430235, 389030773, 680946436, 1192794189

Offset: 0

Ilya Gutkovskiy, Apr 11 2021

nonn

Cf. A001006, A050253, A307971, A343305.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = a[n - 3] + Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 43}]
nmax = 43; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x] + x^4 A[x]^2 + 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) + x^4 * A(x)^2.

A343305 a(0) = ... = a(3) = 1; a(n) = a(n-4) + Sum_{k=0..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 17, 25, 36, 53, 81, 125, 191, 289, 439, 675, 1046, 1621, 2506, 3877, 6023, 9395, 14681, 22947, 35890, 56231, 88285, 138825, 218493, 344145, 542618, 856597, 1353766, 2141383, 3389797, 5370219, 8514773, 13511673, 21456808, 34096503, 54216636

Offset: 0

Ilya Gutkovskiy, Apr 11 2021

nonn

Cf. A001006, A050253, A307972, A343304.

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[n - 4] + Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 45}]
nmax = 45; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x] + x^5 A[x]^2 + 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) + x^5 * A(x)^2.

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

Original entry on oeis.org

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

A108296 Diagonal sums of the number triangle associated to A086617.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 43, 78, 144, 269, 509, 971, 1868, 3618, 7049, 13805, 27162, 53661, 106405, 211697, 422458, 845386, 1696017, 3410522, 6873060, 13878721, 28077439, 56900936, 115501012, 234807488, 478032437, 974507543, 1989123814

Offset: 0

Paul Barry, May 31 2005

The triangle associated to A086617 is given by T(n,k)=if(k<=n, sum{j=0..n-k, C(n-k,j)C(k,j)C(j)},0). A050253(n)=A108296(n+2)-A108296(n).

G.f.: (1-x^2-sqrt(1-2*x^2-4*x^3-3*x^4))/(2*x^3*(1-x^2)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2*k} binomial(n-2*k, j)*binomial(k, j) * A000108(j).
Conjecture: (n+3)*a(n) +(-n-2)*a(n-1) +2*(-n-1)*a(n-2) +2*(-n+3)*a(n-3) +(n+1)*a(n-4) +3*(n-2)*a(n-5)=0. - R. J. Mathar, Nov 16 2012

cp's OEIS Frontend

A343304 a(0) = a(1) = a(2) = 1; a(n) = a(n-3) + Sum_{k=0..n-4} a(k) * a(n-k-4).

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A343305 a(0) = ... = a(3) = 1; a(n) = a(n-4) + Sum_{k=0..n-5} a(k) * a(n-k-5).

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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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A108296 Diagonal sums of the number triangle associated to A086617.

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