A369918 - OEIS

%I #5 Feb 05 2024 13:48:16

%S 0,1,0,3,0,14,6,22,18,46,30,82,42,95,14,109,108,204,148,320,208,389,

%T 184,443,434,761,558,999,502,926,488,986,1020,1840,1204,1917,1388,

%U 1884,990,2416,1870,3029,2172,3895,2060,3466,2146,3938,3562,5395,3708,6269,4214,5800

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e 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.

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

%Y Cf. A003601, A245656, A369916.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Feb 05 2024