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A308025
a(n) = n*(2*n - 3 - (-1)^n)*(5*n - 2 + (-1)^n)/16.
1
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.
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, May 09 2019
STATUS
approved