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 1-7 of 7 results.

A347806 Expansion of (theta_3(x) - 1)^3 / (4 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 4, 5, 6, 10, 14, 22, 30, 41, 62, 88, 123, 173, 248, 354, 500, 710, 1006, 1427, 2024, 2867, 4066, 5767, 8176, 11591, 16436, 23301, 33032, 46832, 66396, 94137, 133461, 189209, 268252, 380315, 539190, 764431, 1083764, 1536498, 2178364, 3088363, 4378502, 6207581
Offset: 3

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

Number of compositions (ordered partitions) of n into 3 or more squares.

Links

Crossrefs

Cf. A000290, A006456, A337165, A347805, A347807, A347808, A347809.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..47);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^3/(4 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 3] &

Formula

a(n) = Sum_{k=3..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

A347807 Expansion of (theta_3(x) - 1)^4 / (8 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 14, 19, 29, 41, 56, 88, 123, 170, 245, 351, 500, 704, 1003, 1427, 2021, 2867, 4060, 5763, 8176, 11585, 16430, 23301, 33032, 46826, 66393, 94131, 133458, 189209, 268243, 380315, 539190, 764422, 1083758, 1536495, 2178361, 3088357, 4378496, 6207581
Offset: 4

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

Number of compositions (ordered partitions) of n into 4 or more squares.

Links

Crossrefs

Cf. A000290, A006456, A337165, A347805, A347806, A347808, A347809.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..47);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^4/(8 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 4] &

Formula

a(n) = Sum_{k=4..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

A347808 Expansion of (theta_3(x) - 1)^5 / (16 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 6, 7, 8, 19, 25, 37, 56, 76, 122, 170, 233, 347, 494, 700, 991, 1415, 2021, 2855, 4054, 5751, 8164, 11585, 16406, 23285, 33032, 46814, 66375, 94119, 133445, 189193, 268231, 380287, 539184, 764422, 1083722, 1536479, 2178349, 3088333, 4378472, 6207557
Offset: 5

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

Number of compositions (ordered partitions) of n into 5 or more squares.

Links

Crossrefs

Cf. A000290, A006456, A337165, A347805, A347806, A347807, A347809.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..47);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^5/(16 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 5] &

Formula

a(n) = Sum_{k=5..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

A347809 Expansion of (theta_3(x) - 1)^6 / (32 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 7, 8, 9, 25, 32, 46, 76, 102, 165, 233, 317, 488, 690, 971, 1395, 1991, 2850, 4024, 5721, 8144, 11550, 16396, 23225, 32987, 46814, 66315, 94069, 133415, 189148, 268181, 380227, 539114, 764387, 1083692, 1536369, 2178299, 3088302, 4378362, 6207477
Offset: 6

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

Number of compositions (ordered partitions) of n into 6 or more squares.

Links

Crossrefs

Cf. A000290, A006456, A337165, A347805, A347806, A347807, A347808.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..47);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^6/(32 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 6] &

Formula

a(n) = Sum_{k=6..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

A348524 Number of compositions (ordered partitions) of n into two or more cubes.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 150, 187, 232, 286, 351, 430, 527, 649, 802, 993, 1230, 1522, 1880, 2318, 2854, 3514, 4330, 5341, 6594, 8145, 10061, 12423, 15330, 18908, 23316, 28753, 35467, 43762
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 21 2021

Keywords

Links

Crossrefs

Cf. A000578, A010057, A023358, A347805, A348522.

Programs

  • Maple
    g:= proc(n) option remember;
     local i,m,t;
     m:= surd(n,3);
     if m::integer then t:= 1; m:= m-1 else t:= 0; m:= floor(m) fi;
     t + add(procname(n-i^3),i=1..m)
end proc:
f:= proc(n) local m;
    m:= surd(n,3);
    if m::integer then g(n)-1 else g(n) fi
end proc:
f(0):= 0:
map(f, [$0..100]);
  • Mathematica
    g[n_] := g[n] = Module[{m, t}, m = n^(1/3); If[IntegerQ[m], t = 1; m = m - 1, t = 0; m = Floor[m]]; t + Sum[g[n - i^3], {i, 1, m}]];
f[n_] := Module[{m}, m = n^(1/3); If[IntegerQ[m], g[n]-1, g[n]]];
f[0] = 0;
Map[f, Range[0, 100]] (* Jean-François Alcover, Sep 19 2022, after Robert Israel *)

A348529 Number of compositions (ordered partitions) of n into two or more triangular numbers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 11, 16, 25, 39, 61, 94, 147, 227, 350, 546, 846, 1309, 2030, 3147, 4875, 7558, 11715, 18154, 28136, 43609, 67586, 104747, 162346, 251610, 389958, 604381, 936699, 1451743, 2249991, 3487152, 5404570, 8376292, 12982016, 20120202, 31183350, 48329596, 74903735
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 21 2021

Keywords

Crossrefs

Cf. A000217, A010054, A023361, A347805, A348526, A348528.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(
     `if`(issqr(8*j+1), b(n-j), 0), j=1..n))
    end:
a:= n-> b(n)-`if`(issqr(8*n+1), 1, 0):
seq(a(n), n=0..43);  # Alois P. Heinz, Oct 21 2021
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, Sum[
     If[IntegerQ@ Sqrt[8*j + 1], b[n - j], 0], {j, 1, n}]];
a[n_] := b[n] - If[IntegerQ@ Sqrt[8*n + 1], 1, 0];
Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

Formula

a(n) = A023361(n) - A010054(n). - Alois P. Heinz, Oct 21 2021

A348326 Number of compositions (ordered partitions) of n into two or more distinct squares.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 8, 0, 0, 8, 30, 0, 0, 0, 2, 6, 0, 2, 6, 24, 2, 8, 6, 0, 0, 8, 30, 0, 0, 6, 32, 24, 2, 8, 30, 120, 6, 24, 2, 6, 0, 8, 36, 24, 0, 34, 150, 0, 2, 12, 30, 24, 0, 2, 38, 150, 0, 12, 78, 144, 2, 30, 122, 6
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 12 2021

Keywords

Examples

			For n = 14 there exists the following six solutions: 1+4+9 = 1+9+4 = 4+1+9 = 4+9+1 = 9+1+4 = 9+4+1 = 14, therefore a(14) = 6. - _Antti Karttunen_, Oct 17 2021

Links

Crossrefs

Cf. A000290, A010052, A033985, A219610, A331844, A347805.
Showing 1-7 of 7 results.

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