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 1-10 of 14 results. Next

A121550 Number of ordered ways of writing n as a sum of three Fibonacci numbers (only one 1 is considered as a Fibonacci number).

Original entry on oeis.org

0, 0, 1, 3, 6, 7, 9, 9, 10, 9, 12, 12, 9, 9, 10, 12, 12, 12, 12, 6, 9, 6, 12, 13, 9, 12, 12, 9, 12, 6, 12, 6, 0, 9, 6, 9, 15, 9, 13, 9, 6, 12, 9, 12, 9, 0, 12, 6, 6, 12, 0, 6, 0, 0, 9, 6, 9, 12, 9, 15, 9, 6, 13, 6, 9, 6, 0, 12, 9, 9, 12, 0, 9, 0, 0, 12, 6, 6, 6, 0, 12, 0, 0, 6, 0, 0, 0, 0, 9, 6, 9, 12
Offset: 1

Views

Author

Emeric Deutsch, Aug 07 2006

Keywords

Examples

			a(6)=7 because we have 6=1+2+3=1+3+2=2+1+3=2+3+1=3+1+2=3+2+1=2+2+2.
		

Crossrefs

Programs

  • Maple
    with(combinat): g:=sum(z^fibonacci(i),i=2..30)^3: gser:=series(g,z=0,130): seq(coeff(gser,z,n),n=1..126);

Formula

G.f.: (Sum_{i>=2} x^Fibonacci(i))^3.
a(n) = A121548(n,3).

A121549 Number of ordered ways of writing n as a sum of two Fibonacci numbers (only one 1 is considered as a Fibonacci number).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 2, 2, 3, 2, 0, 2, 2, 2, 3, 0, 2, 0, 0, 2, 2, 2, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0
Offset: 1

Views

Author

Emeric Deutsch, Aug 07 2006

Keywords

Examples

			a(6)=3 because we have 6=1+5=3+3=5+1.
		

Crossrefs

Programs

  • Maple
    with(combinat): g:=sum(z^fibonacci(i),i=2..30)^2: gser:=series(g,z=0,130): seq(coeff(gser,z,n),n=1..126);

Formula

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

A357688 Number of ways to write n as an ordered sum of four positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 4, 10, 16, 23, 28, 34, 36, 43, 48, 50, 48, 50, 56, 58, 64, 67, 60, 58, 52, 64, 64, 70, 68, 70, 76, 70, 72, 79, 60, 60, 48, 58, 68, 60, 84, 80, 64, 82, 64, 82, 88, 66, 76, 66, 64, 84, 60, 79, 60, 24, 60, 36, 60, 74, 48, 88, 76, 72, 96, 68, 88, 76, 48, 82, 60, 70
Offset: 4

Views

Author

Ilya Gutkovskiy, Oct 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^4.
a(n) = A121548(n,4).

A357690 Number of ways to write n as an ordered sum of five positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 5, 15, 30, 50, 71, 95, 115, 140, 165, 191, 205, 220, 240, 260, 285, 310, 325, 325, 320, 341, 350, 380, 385, 405, 420, 430, 450, 465, 465, 445, 410, 435, 425, 450, 481, 495, 515, 490, 510, 555, 525, 580, 540, 530, 570, 530, 580, 600, 520, 525, 440, 455, 520, 445, 555, 530
Offset: 5

Views

Author

Ilya Gutkovskiy, Oct 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 61; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^5.
a(n) = A121548(n,5).

A357691 Number of ways to write n as an ordered sum of six positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 6, 21, 50, 96, 156, 231, 312, 405, 506, 621, 726, 828, 930, 1041, 1160, 1290, 1422, 1520, 1590, 1677, 1766, 1887, 1980, 2106, 2196, 2310, 2426, 2550, 2670, 2706, 2700, 2736, 2756, 2850, 2916, 3071, 3156, 3186, 3296, 3396, 3510, 3621, 3636, 3765, 3720, 3840, 3966, 4010
Offset: 6

Views

Author

Ilya Gutkovskiy, Oct 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^6.
a(n) = A121548(n,6).

A359514 Number of compositions (ordered partitions) of n into at most 2 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 2, 3, 2, 3, 2, 0, 3, 2, 2, 3, 0, 2, 0, 0, 3, 2, 2, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 3, 2, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 2, 0, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 03 2023

Keywords

Crossrefs

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, 2):
    seq(a(n), n=0..94);  # Alois P. Heinz, Jan 03 2023
  • Mathematica
    g[n_] := IntegerQ@Sqrt[# + 4] || IntegerQ@Sqrt[# - 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, 2];
    Table[a[n], {n, 0, 94}] (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..2} A121548(n,k). - Alois P. Heinz, Jan 03 2023

A357694 Number of ways to write n as an ordered sum of seven positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 7, 28, 77, 168, 308, 504, 750, 1050, 1400, 1813, 2261, 2737, 3227, 3753, 4312, 4921, 5579, 6230, 6832, 7413, 8008, 8652, 9289, 9996, 10654, 11361, 12061, 12853, 13657, 14357, 14924, 15393, 15869, 16408, 16933, 17689, 18319, 18949, 19537, 20244, 21049, 21728
Offset: 7

Views

Author

Ilya Gutkovskiy, Oct 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 49; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^7.
a(n) = A121548(n,7).

A357716 Number of ways to write n as an ordered sum of eight positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 8, 36, 112, 274, 560, 1008, 1640, 2479, 3536, 4844, 6392, 8170, 10136, 12308, 14680, 17291, 20160, 23248, 26440, 29674, 32992, 36456, 40040, 43834, 47712, 51752, 55840, 60250, 64856, 69560, 74088, 78331, 82440, 86500, 90616, 95074, 99568, 104188, 108528, 113304
Offset: 8

Views

Author

Ilya Gutkovskiy, Oct 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

Formula

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^8.
a(n) = A121548(n,8).

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

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

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
Showing 1-10 of 14 results. Next