A348165 -id:A348165 - cp's OEIS frontend

A348892 Number of solutions to +-1^3 +- 2^3 +- 3^3 +- ... +- n^3 = n.

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 4, 0, 0, 83, 69, 0, 0, 353, 414, 0, 0, 7800, 12496, 0, 0, 48162, 56870, 0, 0, 733392, 1253467, 0, 0, 4892337, 10022277, 0, 0, 45859303, 149422926, 0, 0, 623257759, 1339056922, 0, 0, 7453502893, 13446831198

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Cf. A063890, A158118, A348165.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > (i*(i+1)//2)**2: return 0
    if i == 0: return 1
    return b(n+i**3, i-1) + b(abs(n-i**3), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(54)]) # Michael S. Branicky, Jan 28 2022

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

A368243 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n^2.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 5, 15, 0, 0, 127, 184, 0, 0, 819, 1382, 0, 0, 9441, 18176, 0, 0, 96562, 172371, 0, 0, 1192142, 2252342, 0, 0, 13869696, 25741462, 0, 0, 177056022, 334176492, 0, 0, 2207693292, 4182801839, 0, 0, 28966597122, 55125154468

Offset: 0

Ilya Gutkovskiy, Jan 22 2024

nonn

Cf. A063890, A158092, A348165.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^2), i-1) +b(n+i^2, i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n^2, n)):
seq(a(n), n=0..49);  # Alois P. Heinz, Jan 22 2024

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

A369433 a(n) = [x^n] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

Original entry on oeis.org

1, 1, 0, 1, 3, 5, 7, 13, 35, 82, 168, 409, 1035, 2540, 6262, 16068, 41474, 107259, 279256, 736359, 1953946, 5205746, 13938670, 37567522, 101675407, 276158642, 752927255, 2060852216, 5658658210, 15582628517, 43032891276, 119166025289, 330808837377

Offset: 0

Ilya Gutkovskiy, Jan 23 2024

nonn

Cf. A000290, A316706, A348165, A350249.

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

A351002 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = n.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 3, 0, 4, 3, 9, 0, 27, 43, 71, 0, 190, 318, 604, 0, 1846, 3127, 5664, 0, 19048, 34065, 62045, 0, 205713, 378243, 705836, 0, 2403370, 4434940, 8276125, 0, 28980680, 54167797, 101541048, 0, 358095372, 674776903, 1274888645, 0, 4551828850, 8612421500

Offset: 0

Ilya Gutkovskiy, Jan 29 2022

nonn

Cf. A000217, A058498, A063890, A158380, A348165, A350287.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > i*(i+1)*(i+2)//6: return 0
    if i == 0: return 1
    return b(n+i*(i+1)//2, i-1) + b(abs(n-i*(i+1)//2), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(50)]) # Michael S. Branicky, Jan 29 2022

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

A369730 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 2, 0, 0, 5, 2, 0, 0, 13, 43, 0, 0, 193, 274, 0, 0, 1552, 3245, 0, 0, 18628, 31048, 0, 0, 188536, 372710, 0, 0, 2376996, 4197425, 0, 0, 27465147, 53072709, 0, 0, 351329160, 650125358, 0, 0, 4398613111, 8429649875, 0, 0, 57629346805, 108986428106

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A000290, A063866, A158092, A348165, A350403, A369731, A369732.

b:= proc(n, i) option remember; `if`(n>i*(i+1)*(2*i+1)/6, 0,
      `if`(i=0, 1, b(n+i^2, i-1)+b(abs(n-i^2), i-1)))
    end:
a:=n-> b(1, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 30 2024

Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2)), {k, 1, n}], x, 1], {n, 0, 48}]

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

A368206 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 8, 13, 0, 0, 272, 0, 0, 0, 5400, 8915, 0, 0, 30433, 1590, 0, 0, 68638, 73470, 0, 0, 90808, 6638072, 0, 0, 127356, 319803, 0, 0, 20130146, 559282596, 0, 0, 1507066936, 3820244957, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

Cf. A000583, A063890, A158465, A348165, A348892.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 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-> `if`(irem(n, 4)>1, 0, b(n, n)):
seq(a(n), n=0..43);  # Alois P. Heinz, Jan 25 2024

a(46)-a(59) from Alois P. Heinz, Jan 25 2024

cp's OEIS Frontend

A348892 Number of solutions to +-1^3 +- 2^3 +- 3^3 +- ... +- n^3 = n.

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A368243 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n^2.

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Maple

Formula

A369433 a(n) = [x^n] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

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Mathematica

A351002 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = n.

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Author

Keywords

Crossrefs

Programs

Python

Formula

A369730 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = 1.

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Maple

Mathematica

Formula

A368206 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

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Maple

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