cp's OEIS Frontend

A226141 Sum of the squared parts of the partitions of n into exactly two parts.

%I A226141 #28 Jul 16 2025 01:22:18
%S A226141 0,2,5,18,30,64,91,156,204,310,385,542,650,868,1015,1304,1496,1866,
%T A226141 2109,2570,2870,3432,3795,4468,4900,5694,6201,7126,7714,8780,9455,
%U A226141 10672,11440,12818,13685,15234,16206,17936,19019,20940,22140,24262,25585,27918,29370,31924,33511
%N A226141 Sum of the squared parts of the partitions of n into exactly two parts.
%H A226141 Michael De Vlieger, <a href="/A226141/b226141.txt">Table of n, a(n) for n = 1..10000</a>
%H A226141 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A226141 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A226141 a(n) = Sum_{i=1..floor(n/2)} (i^2 + (n-i)^2).
%F A226141 a(n) = n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8. - _Giovanni Resta_, May 29 2013
%F A226141 G.f.: x^2*(2+3*x+7*x^2+3*x^3+x^4) / ( (1+x)^3*(x-1)^4 ). - _R. J. Mathar_, Jun 07 2013
%F A226141 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). - _Wesley Ivan Hurt_, Jun 22 2024
%F A226141 a(n) = A000330(n) - A308422(n). - _Wesley Ivan Hurt_, Jul 16 2025
%e A226141 a(5) = 30; 5 has exactly 2 partitions into two parts, (4,1) and (3,2). Squaring the parts and adding, we get: 1^2 + 2^2 + 3^2 + 4^2 = 30.
%p A226141 a:=n->sum(i^2 + (n-i)^2, i=1..floor(n/2)); seq((a(k), k=1..40);
%t A226141 Array[Sum[i^2 + (# - i)^2, {i, Floor[#/2]}] &, 39] (* _Michael De Vlieger_, Jan 23 2018 *)
%t A226141 LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,2,5,18,30,64,91},50] (* _Harvey P. Dale_, Jul 23 2019 *)
%o A226141 (Magma) [n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8 : n in [1..80]]; // _Wesley Ivan Hurt_, Jun 22 2024
%Y A226141 Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), this sequence (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).
%Y A226141 Cf. A000330, A308422.
%K A226141 nonn,easy
%O A226141 1,2
%A A226141 _Wesley Ivan Hurt_, May 27 2013