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 1-4 of 4 results.

A369734 Number of solutions to 1^2*k_1 + 2^2*k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 8, 17, 35, 79, 177, 409, 995, 2475, 6336, 16078, 41401, 107304, 279550, 736032, 1950000, 5199850, 13950852, 37576658, 101670863, 276228026, 753114256, 2060995699, 5658663542, 15583477334, 43039552072, 119179072495, 330836308272, 920537185436
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A350249, A369433, A369628, A369735.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2)), {k, 1, n}], x, 1], {n, 0, 32}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

A369735 Number of solutions to 1^3*k_1 + 2^3*k_2 + ... + n^3*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 4, 6, 15, 28, 56, 125, 287, 646, 1540, 3625, 8484, 21167, 51458, 126342, 323126, 811538, 2052501, 5339265, 13751212, 35589866, 94032931, 246791641, 650227636, 1739032299, 4630165425, 12373805281, 33429284691, 90073865814, 243460560324
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A369345, A369437, A369628, A369734.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>(i*(i+1)/2)^2, 0,
     `if`(i=0, 1, b(n, i-1)+b(n+i^3, i-1)+b(abs(n-i^3), i-1)))
    end:
a:= n-> b(1, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    Table[Coefficient[Product[(x^(k^3) + 1 + 1/x^(k^3)), {k, 1, n}], x, 1], {n, 0, 33}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).

A369714 Number of solutions to 1^4*k_1 + 2^4*k_2 + ... + n^4*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 13, 25, 56, 110, 218, 494, 1216, 2702, 6477, 14752, 35758, 83730, 208107, 499459, 1250815, 3048590, 7787399, 19260830, 49686365, 124430675, 324018684, 820906005, 2155194085, 5514650519, 14578030389, 37630395887, 100201473164
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A368478, A369358, A369628, A369734, A369735.

Programs

  • Maple
    b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1, b(n, i-1)+
      b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
    end:
a:= n-> b(1, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[n, i-1] + b[Abs[n-i^4], i-1] + b[n+i^4, i-1]]]][i*(i+1)*(2*i+1)*(3*i^2 + 3*i-1)/30];
a[n_] := b[1, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 27 2025, after Alois P. Heinz *)

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Extensions

a(34)-a(37) from Alois P. Heinz, Jan 30 2024

A369733 Number of solutions to 2*k_1 + 3*k_2 + ... + prime(n)*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 0, 1, 1, 3, 8, 18, 39, 95, 233, 565, 1440, 3640, 9409, 24957, 66699, 177931, 475584, 1292985, 3517163, 9636135, 26675682, 73847316, 205379443, 571621138, 1588186858, 4435778209, 12474517743, 35194193531, 99781874834, 283513309423, 799779819641
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A113040, A350880, A369560, A369628.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(1, n):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    Table[Coefficient[Product[(x^Prime[k] + 1 + 1/x^Prime[k]), {k, 1, n}], x, 1], {n, 0, 31}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).
Showing 1-4 of 4 results.

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