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.

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).

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

A346073 a(n) = 1 + Sum_{k=0..n-4} a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 45, 73, 118, 185, 293, 475, 778, 1263, 2047, 3345, 5512, 9085, 14957, 24683, 40918, 67987, 113016, 188053, 313608, 524041, 876657, 1467797, 2460644, 4130893, 6942726, 11678687, 19663068, 33139295, 55904339, 94384167, 159470488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A000108, A007317, A090344, A216604, A307971, A346074.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 42}]
nmax = 42; A[] = 0; Do[A[x] = 1/(1 - x) + x^4 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023
  • SageMath
    @CachedFunction
def a(n): # a = A346073
    if (n<4): return 1
    else: return 1 + sum(a(k)*a(n-k-4) for k in range(n-3))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^2.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 11 2021

Keywords

Crossrefs

Cf. A001006, A050253, A307971, A343305.

Programs

  • Mathematica
    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]

Formula

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

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

Views

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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]

Formula

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

A346076 a(n) = 1 + Sum_{k=1..n-4} a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 17, 25, 36, 54, 84, 131, 201, 307, 475, 745, 1172, 1837, 2878, 4531, 7173, 11381, 18057, 28669, 45624, 72796, 116336, 186066, 297865, 477505, 766621, 1232214, 1982292, 3191693, 5143974, 8298640, 13399691, 21652705, 35014373, 56663700
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A105633, A217282, A307971, A343305, A346048, A346073, A346075, A346077.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 1, n - 4}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = 1/(1 - x) + x^4 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • SageMath
    @CachedFunction
def a(n): # a = A346076
    if (n<5): return 1
    else: return 1 + sum(a(k)*a(n-k-4) for k in range(1,n-3))
[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^4 * A(x) * (A(x) - 1).

A336009 a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 16, 34, 82, 226, 678, 2108, 6892, 23948, 88532, 344816, 1401200, 5925000, 26146360, 120743496, 582606552, 2926675112, 15259183112, 82458502624, 461577781968, 2674216518016, 16013654472352, 98968416103968, 630595248710144
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2020

Keywords

Comments

Shifts 4 places left when e.g.f. is squared.

Crossrefs

Cf. A000142, A007558, A307971, A333497, A336010.

Programs

  • Mathematica
    a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[Binomial[n - 4, k] a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + Integrate[Integrate[Integrate[Integrate[A[x]^2, 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 + Integral( Integral( Integral( Integral A(x)^2 dx) dx) dx) dx.

A346734 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 6, 10, 15, 27, 55, 111, 210, 388, 741, 1473, 2956, 5856, 11514, 22806, 45756, 92394, 186459, 375867, 759519, 1541803, 3140775, 6407307, 13081230, 26745378, 54797850, 112495734, 231270690, 475960278, 980643070, 2023057266, 4178837181, 8641346835
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

Crossrefs

Cf. A001764, A019497, A307971, A346733, A346735.

Programs

  • Mathematica
    nmax = 37; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 4, 1, Sum[Sum[a[i] a[j] a[n - i - j - 4], {j, 0, n - i - 4}], {i, 0, n - 4}]]; Table[a[n], {n, 0, 37}]

Formula

a(0) = ... = a(3) = 1; a(n) = Sum_{i=0..n-4} Sum_{j=0..n-i-4} a(i) * a(j) * a(n-i-j-4).
Showing 1-8 of 8 results.

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