A302766 a(n) = n*((4*n + 1)*(7*n - 4) + 15*n*(-1)^n)/48.

0, 5, 11, 39, 60, 130, 175, 306, 384, 595, 715, 1025, 1196, 1624, 1855, 2420, 2720, 3441, 3819, 4715, 5180, 6270, 6831, 8134, 8800, 10335, 11115, 12901, 13804, 15860, 16895, 19240, 20416, 23069, 24395, 27375, 28860, 32186, 33839, 37530, 39360, 43435, 45451

Offset: 1

Author

Wesley Ivan Hurt, Apr 12 2018

Keywords

• nonn
• easy

Comments

Consider the partitions of n into two parts (p,q). Then 2*a(n) represents the total surface area of the family of rectangular prisms with dimensions p, q and (p + q).

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. A302647, A302758.

Programs

GAP

List([1..50], n -> n*((4*n+1)*(7*n-4)+15*n*(-1)^n)/48); # Bruno Berselli, Apr 16 2018

Magma

[Floor(n/2)*(6*n^2+3*n-1+3*(n-1)*Floor(n/2)-2*Floor(n/2)^2)/6 : n in [1..45]]; // Vincenzo Librandi, Apr 13 2018

Mathematica

Table[Floor[n/2] (6 n^2 + 3 n - 1 + 3 (n - 1) Floor[n/2] - 2 Floor[n/2]^2)/6, {n, 50}]

PARI

a(n) = floor(n/2)*(6*n^2+3*n-1+3*(n-1)*floor(n/2)-2*floor(n/2)^2)/6 \\ Felix Fröhlich, Apr 13 2018

PARI

concat(0, Vec(x^2*(5 + 6*x + 13*x^2 + 3*x^3 + x^4) / ((1 - x)^4*(1 + x)^3) + O(x^60))) \\ Colin Barker, Apr 13 2018

Formula

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

a(n) = floor(n/2)*(6*n^2+3*n-1+3*(n-1)*floor(n/2)-2*floor(n/2)^2)/6.

From Colin Barker, Apr 13 2018: (Start)

G.f.: x^2*(5 + 6*x + 13*x^2 + 3*x^3 + x^4) / ((1 - x)^4*(1 + x)^3).

a(n) = (14*n^3 + 3*n^2 - 2*n) / 24 for n even.

a(n) = (14*n^3 - 12*n^2 - 2*n) / 24 for n odd.

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)