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

Author

Wesley Ivan Hurt, May 09 2019

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A294286.

Programs

Mathematica

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

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

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