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.

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A359515 Number of compositions (ordered partitions) of n into at most 3 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 10, 11, 12, 12, 12, 14, 12, 12, 11, 12, 15, 12, 14, 12, 6, 12, 8, 14, 15, 9, 15, 12, 9, 14, 6, 12, 6, 0, 12, 8, 11, 17, 9, 15, 9, 6, 15, 9, 12, 9, 0, 14, 6, 6, 12, 0, 6, 0, 0, 12, 8, 11, 14, 9, 17, 9, 6, 15, 6, 9, 6, 0, 15, 9, 9, 12, 0, 9, 0, 0, 14, 6, 6, 6, 0, 12
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 03 2023

Keywords

Crossrefs

Cf. A000045, A076739, A121548, A121550, A359512, A359514, A359516.

Programs

  • Maple
    g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
b:= proc(n, t) option remember; `if`(n=0, 1, `if`(t<1, 0,
      add(`if`(g(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..81);  # Alois P. Heinz, Jan 03 2023
  • Mathematica
    g[n_] := With[{t = 5 n^2}, IntegerQ @ Sqrt[t+4] || IntegerQ @ Sqrt[t-4]];
b[n_, t_] := b[n, t] = If[n == 0, 1, If[t < 1, 0, Sum[If[g[j], b[n-j, t-1], 0], {j, 1, n}]]];
a[n_] :=  b[n, 3];
Table[a[n], {n, 0, 81}] (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..3} A121548(n,k). - Alois P. Heinz, Jan 03 2023

A359516 Number of compositions (ordered partitions) of n into at most 4 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 20, 27, 35, 40, 46, 50, 55, 60, 61, 60, 65, 68, 72, 76, 73, 72, 66, 66, 79, 73, 85, 80, 79, 90, 76, 84, 85, 60, 72, 56, 69, 85, 69, 99, 89, 70, 97, 73, 94, 97, 66, 90, 72, 70, 96, 60, 85, 60, 24, 72, 44, 71, 88, 57, 105, 85, 78, 111, 74, 97, 82, 48, 97, 69, 79
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 03 2023

Keywords

Crossrefs

Cf. A000045, A076739, A121548, A357688, A359513, A359514, A359515.

Programs

  • Maple
    g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
b:= proc(n, t) option remember; `if`(n=0, 1, `if`(t<1, 0,
      add(`if`(g(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 03 2023
  • Mathematica
    g[n_] := Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2];
b[n_, t_] := b[n, t] = If[n == 0, 1, If[t < 1, 0, Sum[If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[n, 4];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..4} A121548(n,k). - Alois P. Heinz, Jan 03 2023

A080888 Number of compositions into Fibonacci numbers (1 counted as two distinct Fibonacci numbers).

Original entry on oeis.org

1, 2, 5, 13, 33, 85, 218, 559, 1435, 3682, 9448, 24244, 62210, 159633, 409622, 1051099, 2697145, 6920936, 17759282, 45570729, 116935544, 300059313, 769959141, 1975732973, 5069776531, 13009163899, 33381815615, 85658511370, 219801722429, 564016306267
Offset: 0

Views

Author

Vladeta Jovovic, Mar 30 2003

Keywords

Examples

			a(2) = 5 since 2 = 1+1 = 1+1' = 1'+1 = 1'+1'.

Links

Crossrefs

Cf. A000045, A076739.

Programs

  • Maple
    a:= proc(n) option remember; local r, f;
      if n=0 then 1 else r, f:= 0, [0, 1];
        while f[2] <= n do r:= r+a(n-f[2]);
          f:= [f[2], f[1]+f[2]]
        od; r
      fi
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 20 2017
  • Mathematica
    a[n_] := a[n] = Module[{r, f}, If[n == 0, 1, {r, f} = {0, {0, 1}}; While[f[[2]] <= n, r = r + a[n - f[[2]]]; f = {f[[2]], f[[1]] + f[[2]]}]; r]];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

Formula

G.f.: 1/(1-Sum_{k>0} x^Fibonacci(k)).
a(n) ~ c * d^n, where d=2.5660231413698319379867000009313373339800958659676443846860312096..., c=0.7633701399876743973524738479037760170533154734693438061127686049... - Vaclav Kotesovec, May 01 2014

A357453 Number of compositions (ordered partitions) of n into tetranacci numbers 1,2,4,8,15,29, ... (A000078).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 31, 56, 98, 174, 306, 542, 956, 1690, 2984, 5273, 9313, 16453, 29062, 51340, 90689, 160203, 282994, 499908, 883078, 1559948, 2755624, 4867776, 8598858, 15189770, 26832521, 47399291, 83730207, 147908288, 261277998, 461544073
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 29 2022

Keywords

Crossrefs

Cf. A000078, A076739, A287656, A357451, A357452, A357455.

Programs

Formula

G.f.: 1 / (1 - Sum_{k>=4} x^A000078(k)).

A357455 Number of compositions (ordered partitions) of n into pentanacci numbers 1,2,4,8,16,31, ... (A001591).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 31, 56, 98, 174, 306, 542, 956, 1690, 2983, 5272, 9310, 16448, 29050, 51318, 90644, 160118, 282826, 499590, 882468, 1558798, 2753448, 4863696, 8591212, 15175514, 26805984, 47350057, 83639033, 147739853, 260967374, 460972308, 814260589
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 29 2022

Keywords

Crossrefs

Cf. A001591, A076739, A288120, A357451, A357453, A357454.

Programs

Formula

G.f.: 1 / (1 - Sum_{k>=5} x^A001591(k)).

A357717 Number of ways to write n as an ordered sum of nine positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 9, 45, 156, 423, 954, 1878, 3321, 5409, 8251, 11979, 16686, 22446, 29250, 37134, 46107, 56259, 67671, 80407, 94338, 109269, 125118, 141930, 159723, 178608, 198522, 219510, 241338, 264438, 288810, 314550, 341010, 367785, 394596, 421443, 448650, 476614, 505404, 534978
Offset: 9

Views

Author

Ilya Gutkovskiy, Oct 10 2022

Keywords

Crossrefs

Cf. A000045, A076739, A121548, A121549, A121550, A319402, A357688, A357690, A357691, A357694, A357716.

Programs

  • Mathematica
    nmax = 47; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^9.
a(n) = A121548(n,9).

A357451 Number of compositions (ordered partitions) of n into tribonacci numbers 1,2,4,7,13,24, ... (A000073).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 32, 57, 101, 179, 318, 564, 1002, 1778, 3157, 5603, 9947, 17656, 31342, 55635, 98759, 175308, 311191, 552400, 980571, 1740625, 3089803, 5484750, 9736045, 17282576, 30678512, 54457808, 96668726, 171597851, 304605465, 540708924
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 29 2022

Keywords

Crossrefs

Cf. A000073, A076739, A117546, A240844, A357453, A357455.

Programs

Formula

G.f.: 1 / (1 - Sum_{k>=3} x^A000073(k)).

A355805 Number of compositions (ordered partitions) of n into Pell numbers (A000129).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 128, 218, 373, 636, 1086, 1853, 3162, 5397, 9210, 15719, 26826, 45782, 78133, 133343, 227568, 388373, 662809, 1131168, 1930482, 3294616, 5622682, 9595828, 16376507, 27948604, 47697869, 81402513, 138923804, 237091241
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A000129, A076739, A290807, A290808.

Programs

  • Mathematica
    nmax = 37; CoefficientList[Series[1/(1 - Sum[x^Fibonacci[k, 2], {k, 1, 20}]), {x, 0, nmax}], x]

Formula

G.f.: 1 / (1 - Sum_{k>=1} x^A000129(k)).

A357519 Number of compositions (ordered partitions) of n into Jacobsthal numbers 1,3,5,11,21,43, ... (A001045).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 12, 19, 30, 47, 75, 118, 185, 292, 460, 725, 1143, 1800, 2836, 4469, 7042, 11097, 17485, 27550, 43411, 68403, 107783, 169834, 267606, 421666, 664419, 1046925, 1649640, 2599335, 4095768, 6453698, 10169086, 16023420, 25248087, 39783383
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A001045, A076739, A296371.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - Sum[x^((2^k - (-1)^k)/3), {k, 2, 20}]), {x, 0, nmax}], x]

Formula

G.f.: 1 / (1 - Sum_{k>=2} x^A001045(k)).

A357730 Number of ways to write n as an ordered sum of ten positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 10, 55, 210, 625, 1542, 3300, 6310, 11040, 17980, 27673, 40660, 57475, 78520, 104175, 134742, 170620, 212220, 260035, 314290, 374933, 441790, 514855, 594210, 680070, 772582, 871920, 977790, 1090680, 1210960, 1339417, 1475340, 1618020, 1766080, 1918785, 2076012
Offset: 10

Views

Author

Ilya Gutkovskiy, Oct 11 2022

Keywords

Crossrefs

Cf. A000045, A076739, A121548, A121549, A121550, A319403, A357688, A357690, A357691, A357694, A357716, A357717.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^10.
a(n) = A121548(n,10).
Previous Showing 11-20 of 21 results. Next

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