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 1-3 of 3 results.

A345456 a(n) = Sum_{k=0..n} binomial(5*n+2,5*k).

Original entry on oeis.org

1, 22, 859, 25773, 843756, 26789257, 859595529, 27481113638, 879683351911, 28146676447417, 900729032983924, 28822936611339453, 922338323835136341, 29514778095285204502, 944473434343229560419, 30223143962480773595093, 967140672636207153780796
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A070782 (b=5,c=0), A345455 (b=5,c=1), this sequence (b=5,c=2), A345457 (b=5,c=3), A345458 (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 2, 5*k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 20 2021 *)
LinearRecurrence[{21,353,-32},{1,22,859},20] (* Harvey P. Dale, Aug 25 2022 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+2, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+x+44*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + x + 44*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+2).
a(n) = 2^(5*n + 3)/10 + ((-295 + 131*sqrt(5))/phi^(5*n) + (115 - 49*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A345457 a(n) = Sum_{k=0..n} binomial(5*n+3,5*k).

Original entry on oeis.org

1, 57, 1574, 53143, 1669801, 53774932, 1717012749, 54986385093, 1759098789526, 56296324109907, 1801425114687749, 57646238657975068, 1844672594930734801, 59029601136140621857, 1888946370232447241574, 60446293452901248074943
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A090408 (b=4,c=3), A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), this sequence (b=5,c=3), A345458 (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 3, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+3, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+36*x+24*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + 36*x + 24*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+3).
a(n) = 2^(5*n + 4)/10 + (( 475 - 213*sqrt(5))/phi^(5*n) + ( 65 - 33*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A345458 a(n) = Sum_{k=0..n} binomial(5*n+4,5*k).

Original entry on oeis.org

1, 127, 3004, 107883, 3321891, 107746282, 3431847189, 109996928003, 3517929664756, 112595619434887, 3602817278095399, 115292842751246298, 3689341137121931721, 118059247217851456567, 3777892242010882603884, 120892592433742197034643
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Links

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), A345457 (b=5,c=3), this sequence (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 4, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
Total/@Table[Binomial[5n+4,5k],{n,0,20},{k,0,n}] (* or *) LinearRecurrence[{21,353,-32},{1,127,3004},30] (* Harvey P. Dale, Oct 29 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+4, 5*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+106*x-16*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + 106*x - 16*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+4).
a(n) = 2^(5*n + 5)/10 + ((2015 - 901*sqrt(5))/phi^(5*n) - (35 + sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021
Showing 1-3 of 3 results.

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