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.

A347805 Expansion of (theta_3(x) - 1)^2 / (2 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 7, 10, 16, 22, 30, 43, 62, 88, 123, 175, 249, 354, 502, 710, 1006, 1427, 2024, 2869, 4068, 5767, 8176, 11593, 16436, 23301, 33033, 46832, 66398, 94137, 133461, 189211, 268252, 380315, 539192, 764433, 1083764, 1536498, 2178364, 3088365, 4378502, 6207581, 8800750
Offset: 2

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A000290, A006456, A010052, A033985, A219610, A337165, A347806, 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, 2):
seq(a(n), n=2..48);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 48; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^2/(2 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 2] &

Formula

From Alois P. Heinz, Sep 14 2021: (Start)
a(n) = A006456(n) - A010052(n).
a(n) = Sum_{k=2..n} A337165(n,k). (End)

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

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
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.