A033985 -id:A033985 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Sep 14 2021

nonn

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

Index entries for sequences related to sums of squares

Cf. A000290, A006456, A010052, A033985, A219610, A337165, A347806, A347807, A347808, A347809.

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

nmax = 48; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^2/(2 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 2] &

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

A348528 Number of partitions of n into two or more distinct triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 3, 1, 1, 2, 2, 1, 4, 3, 0, 3, 3, 1, 4, 3, 3, 3, 2, 3, 3, 2, 3, 6, 4, 2, 5, 4, 2, 6, 5, 3, 6, 6, 3, 5, 5, 5, 6, 5, 4, 7, 7, 5, 8, 6, 5, 9, 7, 4, 9, 9, 6, 10, 9, 3, 9, 10, 8, 11, 11, 9, 10, 10, 9, 10, 10, 9

Offset: 0

Ilya Gutkovskiy, Oct 21 2021

nonn

Cf. A000217, A010054, A024940, A033985, A348526, A348529.

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

Ilya Gutkovskiy, Oct 12 2021

nonn

			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

Index entries for sequences related to sums of squares

Cf. A000290, A010052, A033985, A219610, A331844, A347805.

cp's OEIS Frontend

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

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A348528 Number of partitions of n into two or more distinct triangular numbers.

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

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