A294286 - OEIS

%I #34 Dec 03 2023 10:14:26

%S 0,0,5,10,30,46,91,124,204,260,385,470,650,770,1015,1176,1496,1704,

%T 2109,2370,2870,3190,3795,4180,4900,5356,6201,6734,7714,8330,9455,

%U 10160,11440,12240,13685,14586,16206,17214,19019,20140,22140,23380,25585,26950,29370

%N Sum of the squares of the parts in the partitions of n into two distinct parts.

%H Iain Fox, <a href="/A294286/b294286.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Colin Barker)

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

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

%F From _David A. Corneth_, Oct 27 2017: (Start)

%F For odd n, a(n) = n^3/3 - n^2/2 + n/6 = A000330(n + 1).

%F For even n, a(n) = n^3/3 - 3*n^2/4 + n/6.

%F (End)

%F From _Colin Barker_, Nov 04 2017: (Start)

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

%F 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.

%F (End)

%F a(n) = n*(2*n^2-3*n*(5+(-1)^n)/4+1)/6. - _Wesley Ivan Hurt_, Dec 03 2023

%e For n = 6, there are two ways of partitioning 6 into two distinct parts: 6 = 1+5 and 6 = 2+4.  So a(6) = 1^2 + 5^2 + 2^2 + 4^2 = 46.

%e For n = 7, there are three ways of partitioning 7 into two distinct parts: 7 = 1+6, 7 = 2+5, and 7 = 3+4.  So a(7) = 1^2 + 6^2 + 2^2 + 5^2 + 3^2 + 4^2 = 91. - _Michael B. Porter_, Nov 05 2017

%t Table[Sum[i^2 + (n - i)^2, {i, Floor[(n-1)/2]}], {n, 40}]

%t Table[Total[Flatten[Select[IntegerPartitions[n,{2}],#[[1]]!=#[[2]]&]]^2],{n,50}] (* _Harvey P. Dale_, Dec 02 2022 *)

%o (PARI) first(n) = my(res = vector(n, i, i^3 / 3 - i^2 / 2 + i / 6)); forstep(i = 2, n, 2, res[i] -= i^2 >> 2); res \\ _David A. Corneth_, Oct 27 2017

%o (PARI) concat(vector(2), Vec(x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3) + O(x^60))) \\ _Colin Barker_, Nov 04 2017

%o (Magma) [n*(2*n^2-3*n*(5+(-1)^n)/4+1)/6 : n in [1..60]]; // _Wesley Ivan Hurt_, Dec 03 2023

%Y Cf. A000330.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Oct 26 2017