cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A307971 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4*A(x)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 44, 70, 112, 175, 272, 430, 690, 1107, 1766, 2822, 4542, 7347, 11886, 19222, 31150, 50647, 82518, 134542, 219542, 358808, 587430, 962898, 1579686, 2593967, 4264292, 7017800, 11559548, 19055420, 31437318, 51908076, 85775954, 141841207
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Shifts 4 places left when convolved with itself.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 13*x^10 + ...

Links

Crossrefs

Cf. A000108, A007477, A307970, A307972.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, 1,
      add(a(j)*a(n-4-j), j=0..n-4))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019
  • Mathematica
    terms = 44; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; a[0] = a[1] = a[2] = a[3] = 1; Table[a[n], {n, 0, 44}]
CoefficientList[Series[(1 - Sqrt[1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7])/(2*x^4), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

Formula

G.f.: 1/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - ...)))))))))), a continued fraction.
Recurrence: a(n+4) = Sum_{k=0..n} a(k)*a(n-k).
G.f.: (1 - sqrt(1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7))/(2*x^4). - Vaclav Kotesovec, Sep 27 2023

A307970 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3*A(x)^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 56, 96, 165, 290, 512, 902, 1601, 2862, 5124, 9198, 16585, 29990, 54336, 98702, 179742, 327942, 599432, 1097756, 2013737, 3699596, 6806866, 12541518, 23137270, 42736850, 79031394, 146309968, 271142469, 502978944, 933921458, 1735634266
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Shifts 3 places left when convolved with itself.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 12*x^8 + 19*x^9 + 32*x^10 + ...

Links

Crossrefs

Cf. A000108, A007477, A307971, A307972.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, 1,
      add(a(j)*a(n-3-j), j=0..n-3))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019
  • Mathematica
    terms = 40; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 40}]

Formula

G.f.: 1/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - ...)))))))))), a continued fraction.
Recurrence: a(n+3) = Sum_{k=0..n} a(k)*a(n-k).
a(n) ~ sqrt(3 + 4*r^4 + 8*r^5) / (4*sqrt(Pi)*n^(3/2)*r^(n+3)), where r = 0.51899425841331458784223152875297289010563957455264491744143... is the root of the equation 1 + r + r^2 = 1/(4*r^3). - Vaclav Kotesovec, Jul 03 2021

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

Views

Author

Ilya Gutkovskiy, Apr 11 2021

Keywords

Crossrefs

Cf. A001006, A050253, A307972, A343304.

Programs

  • Mathematica
    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]

Formula

G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x) + x^5 * A(x)^2.

A346049 a(0) = ... = a(4) = 1; a(n) = Sum_{k=1..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 3, 4, 4, 4, 6, 10, 16, 24, 30, 37, 50, 74, 116, 175, 245, 332, 456, 654, 981, 1471, 2146, 3056, 4320, 6203, 9119, 13540, 19986, 29134, 42113, 61047, 89398, 132021, 195272, 287547, 421235, 616418, 905161, 1335648, 1976407, 2922982, 4313230
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A025250, A307972, A343305, A346047, A346048.

Programs

  • Mathematica
    a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[a[k] a[n - k - 5], {k, 1, n - 5}]; Table[a[n], {n, 0, 48}]
nmax = 48; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • SageMath
    @CachedFunction
def a(n): # a = A346049
    if (n<5): return 1
    else: return 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 28 2022

Formula

G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x) * (A(x) - 1).

A346074 a(n) = 1 + 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, 6, 9, 14, 21, 30, 41, 59, 89, 136, 205, 301, 443, 664, 1011, 1545, 2341, 3530, 5341, 8143, 12487, 19148, 29299, 44817, 68721, 105742, 163025, 251392, 387595, 597988, 924047, 1430167, 2215595, 3433788, 5323915, 8260652, 12829849
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A007317, A090344, A216604, A307972, A346073.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x)^2.
Conjecture D-finite with recurrence (n+5)*a(n) +2*(-n-4)*a(n-1) +(n+3)*a(n-2) +2*(-2*n+5)*a(n-5) +4*(n-3)*a(n-6)=0. - R. J. Mathar, Feb 17 2022
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

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

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • SageMath
    @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

Formula

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

A336010 a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} binomial(n-5,k) * a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 66, 148, 374, 1052, 3156, 9724, 31096, 104124, 366696, 1355624, 5220120, 20763160, 84944720, 357759200, 1557192440, 7029575320, 32929457880, 159764303320, 800509163360, 4132518624560, 21953331512080, 119966645509440
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2020

Keywords

Comments

Shifts 5 places left when e.g.f. is squared.

Crossrefs

Cf. A000142, A007558, A307972, A333497, A336009.

Programs

  • Mathematica
    a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[Binomial[n - 5, k] a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integrate[Integrate[Integrate[Integrate[Integrate[A[x]^2, x], x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

Formula

E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integral( Integral( Integral( Integral( Integral A(x)^2 dx) dx) dx) dx) dx.

A346735 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 6, 10, 15, 21, 34, 63, 120, 220, 381, 642, 1102, 1968, 3615, 6658, 12090, 21675, 38820, 70200, 128466, 236583, 435453, 798798, 1462933, 2684352, 4945740, 9145839, 16942356, 31388571, 58140726, 107753364, 199993359, 371852269, 692375844, 1290252474
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

Crossrefs

Cf. A001764, A019497, A307972, A346733, A346734.

Programs

  • Mathematica
    nmax = 40; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 5, 1, Sum[Sum[a[i] a[j] a[n - i - j - 5], {j, 0, n - i - 5}], {i, 0, n - 5}]]; Table[a[n], {n, 0, 40}]

Formula

a(0) = ... = a(4) = 1; a(n) = Sum_{i=0..n-5} Sum_{j=0..n-i-5} a(i) * a(j) * a(n-i-j-5).
Showing 1-8 of 8 results.

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