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 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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, May 09 2019
STATUS
approved