A308038 a(n) = Sum_{i=1..floor((n-1)/2)} i * (n-i)^2.

0, 0, 4, 9, 34, 57, 134, 196, 370, 500, 830, 1065, 1624, 2009, 2884, 3472, 4764, 5616, 7440, 8625, 11110, 12705, 15994, 18084, 22334, 25012, 30394, 33761, 40460, 44625, 52840, 57920, 67864, 73984, 85884, 93177, 107274, 115881, 132430, 142500, 161770, 173460

Offset

1,3

Comments

Total area of all trapezoids with height s*t, and bases s+t and t-s for positive integers s and t, n = s + t, and s < t.

Total volume of all rectangular prisms with dimensions s X t X t, for positive integers s and t such that n = s + t and s < t.

Also, total area of all rectangles with dimensions s X t^2, where s and t are positive integers, n = s + t and s < t.

Consider the partitions of n into two distinct parts (s,t) with s < t. Then a(n) is the sum of all the products s*t^2, using the corresponding parts from each (s,t).

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 (1,4,-4,-6,6,4,-4,-1,1).

Formula

a(n) = (2*n+1-(-1)^n)*(2*n-3-(-1)^n)*(3*(-1+(-1)^n)*(3+(-1)^n)+4*n*(5+5*(-1)^n+11*n))/3072.

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).

From Colin Barker, Jun 22 2019: (Start)

G.f.: x^3*(4 + 5*x + 9*x^2 + 3*x^3 + x^4) / ((1 - x)^5*(1 + x)^4).

a(n) = (3 - 3*(-1)^n - 2*(17+3*(-1)^n)*n^2 - 12*(1+(-1)^n)*n^3 + 22*n^4) / 384.

(End)

Mathematica

Table[Sum[i*(n - i)^2, {i, Floor[(n - 1)/2]}], {n, 60}]

Prog

(PARI) concat([0, 0], Vec(x^3*(4 + 5*x + 9*x^2 + 3*x^3 + x^4) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Jun 22 2019

Crossrefs

Sequence in context: A054433 A219769 A379407 * A096531 A149121 A149122

Adjacent sequences: A308035 A308036 A308037 * A308039 A308040 A308041

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 10 2019

Status

approved