A308025 a(n) = n*(2*n - 3 - (-1)^n)*(5*n - 2 + (-1)^n)/16.

0, 0, 9, 19, 55, 87, 168, 234, 378, 490, 715, 885, 1209, 1449, 1890, 2212, 2788, 3204, 3933, 4455, 5355, 5995, 7084, 7854, 9150, 10062, 11583, 12649, 14413, 15645, 17670, 19080, 21384, 22984, 25585, 27387, 30303, 32319, 35568, 37810, 41410, 43890, 47859

Offset

1,3

Comments

Consider the rectangular prisms with dimensions s X t X t, where n = s + t and s < t. Then a(n) is the sum of the areas of the squares that rest on a given space diagonal in each of the rectangular prisms.

Sum of the squares of the smaller parts and twice the sum of the squares of the larger parts in the partitions of n into two distinct parts.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^2 + 2*(n-i)^2.

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).

G.f.: x^3*(9 + 10*x + 9*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3). - Colin Barker, May 17 2019

Mathematica

Table[n*(2 n - 3 - (-1)^n)*(5 n - 2 + (-1)^n)/16, {n, 60}]

Prog

(PARI) concat([0, 0], Vec(x^3*(9 + 10*x + 9*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, May 17 2019

Crossrefs

Cf. A294286.

Sequence in context: A048696 A046103 A146459 * A360429 A041158 A146080

Adjacent sequences: A308022 A308023 A308024 * A308026 A308027 A308028

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 09 2019

Status

approved