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 1-6 of 6 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

A307972 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5*A(x)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 58, 86, 130, 195, 286, 416, 612, 915, 1380, 2076, 3102, 4627, 6932, 10452, 15818, 23931, 36148, 54600, 82642, 125435, 190724, 290116, 441282, 671512, 1023052, 1560780, 2383578, 3642117, 5567202, 8514254, 13031192, 19960712
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2019

Keywords

Comments

Shifts 5 places left when convolved with itself.

Examples

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

Links

Crossrefs

Cf. A000108, A007477, A307970, A307971.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<5, 1,
      add(a(j)*a(n-5-j), j=0..n-5))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2019
  • Mathematica
    terms = 47; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; a[0] = a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 0, 47}]
  • SageMath
    @CachedFunction
def a(n): # a = A307972
    if (n<5): return 1
    else: return sum(a(k)*a(n-k-5) for k in range(n-4))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

Formula

G.f.: 1/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - x/(1 - x^5/(1 - x^5/(1 - ...)))))))))), a continued fraction.
Recurrence: a(n+5) = Sum_{k=0..n} a(k)*a(n-k).

A333497 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k) * a(n-k-3).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 18, 48, 144, 456, 1560, 5808, 23184, 98160, 440832, 2101824, 10588608, 56104128, 312013440, 1818498816, 11082682368, 70467474816, 466680045312, 3214497245184, 22994283345408, 170573216656896, 1310482565462016, 10415453732637696
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2020

Keywords

Comments

Shifts 3 places left when e.g.f. is squared.

Links

Crossrefs

Cf. A000142, A007558, A307970, A336009, A336010.

Programs

  • Mathematica
    a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = 1 + x + x^2/2 + Integrate[Integrate[Integrate[A[x]^2, 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 + Integral( Integral( Integral A(x)^2 dx) dx) dx.

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

Views

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    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]

Formula

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

A346075 a(n) = 1 + Sum_{k=1..n-3} a(k) * a(n-k-3).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 10, 16, 25, 41, 69, 115, 192, 326, 558, 955, 1641, 2839, 4930, 8578, 14972, 26222, 46037, 80988, 142793, 252307, 446617, 791885, 1406394, 2501642, 4456080, 7947963, 14194221, 25379751, 45430710, 81409233, 146028788, 262192876, 471193406
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A105633, A216604, A217282, A307970, A343304, A346047, A346076, A346077.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 3], {k, 1, n - 3}]; Table[a[n], {n, 0, 40}]
nmax = 40; A[] = 0; Do[A[x] = 1/(1 - x) + x^3 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • SageMath
    @CachedFunction
def a(n): # a = A346075
    if (n<4): return 1
    else: return 1 + sum(a(k)*a(n-k-3) for k in range(1,n-2))
[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^3 * A(x) * (A(x) - 1).

A346733 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 10, 21, 48, 103, 219, 489, 1114, 2517, 5712, 13152, 30492, 70812, 165165, 387456, 912378, 2154250, 5102343, 12123027, 28878384, 68947041, 164979006, 395604531, 950428335, 2287387152, 5514240673, 13314167718, 32194109193, 77953239507, 188997294360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

Crossrefs

Cf. A001764, A019497, A307970, A346734, A346735.

Programs

  • Mathematica
    nmax = 34; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Sum[a[i] a[j] a[n - i - j - 3], {j, 0, n - i - 3}], {i, 0, n - 3}]]; Table[a[n], {n, 0, 34}]

Formula

a(0) = a(1) = a(2) = 1; a(n) = Sum_{i=0..n-3} Sum_{j=0..n-i-3} a(i) * a(j) * a(n-i-j-3).
Showing 1-6 of 6 results.

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