A369918 Total area of all distinct rectangles with semiperimeter n whose side lengths are arithmetic numbers.

0, 1, 0, 3, 0, 14, 6, 22, 18, 46, 30, 82, 42, 95, 14, 109, 108, 204, 148, 320, 208, 389, 184, 443, 434, 761, 558, 999, 502, 926, 488, 986, 1020, 1840, 1204, 1917, 1388, 1884, 990, 2416, 1870, 3029, 2172, 3895, 2060, 3466, 2146, 3938, 3562, 5395, 3708, 6269, 4214, 5800

Offset

1,4

Links

Table of n, a(n) for n=1..54.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * c(k) * c(n-k), where c = A245656.

Example

a(14) = 95. There are 3 distinct rectangles with semiperimeter 14 whose side lengths are arithmetic numbers. They are: 13 X 1, 11 X 3, and 7 X 7. The total area of the rectangles is 13*1 + 11*3 + 7*7 = 95.

Mathematica

a[n_] := Boole[IntegerQ[Mean[Divisors[n]]]]; Table[Sum[k*(n - k) a[k]*a[n - k], {k, Floor[n/2]}], {n, 100}]

Crossrefs

Cf. A003601, A245656, A369916.

Sequence in context: A350826 A008403 A138349 * A323689 A321413 A135399

Adjacent sequences: A369915 A369916 A369917 * A369919 A369920 A369921

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 05 2024

Status

approved