A298992 a(n) = (2*n-3-(-1)^n)*(22*n^2-21*n+5*n*(-1)^n)/96.

0, 0, 5, 12, 35, 58, 112, 160, 258, 340, 495, 620, 845, 1022, 1330, 1568, 1972, 2280, 2793, 3180, 3815, 4290, 5060, 5632, 6550, 7228, 8307, 9100, 10353, 11270, 12710, 13760, 15400, 16592, 18445, 19788, 21867, 23370, 25688, 27360, 29930, 31780, 34615, 36652

Offset

1,3

Comments

Consider the partitions of n into two distinct parts (p,q) where p < q. Then a(n) is the total area of the family of rectangles (and the areas of the squares on their sides) with dimensions p and |q - p|.

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*(n-2*i) + 2*i^2 + 2*(n-2*i)^2.

From Colin Barker, Apr 23 2019: (Start)

G.f.: x^3*(5 + 7*x + 8*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3).

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) for n>7.

(End)

Mathematica

Table[(2 n - 3 - (-1)^n) (22 n^2 - 21 n + 5 n (-1)^n)/96, {n, 50}]

Prog

(PARI) concat([0, 0], Vec(x^3*(5 + 7*x + 8*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, Apr 23 2019

Crossrefs

Cf. A302647, A302758.

Sequence in context: A192243 A292104 A136113 * A050189 A308344 A116995

Adjacent sequences: A298989 A298990 A298991 * A298993 A298994 A298995

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 16 2018

Status

approved