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A110660 Oblong (promic) numbers repeated.

%I A110660 #61 Feb 16 2025 08:32:58
%S A110660 0,0,2,2,6,6,12,12,20,20,30,30,42,42,56,56,72,72,90,90,110,110,132,
%T A110660 132,156,156,182,182,210,210,240,240,272,272,306,306,342,342,380,380,
%U A110660 420,420,462,462,506,506,552,552,600,600,650,650,702,702,756,756,812,812
%N A110660 Oblong (promic) numbers repeated.
%C A110660 a(floor(n/2)) = A002378(n).
%C A110660 Sum of the even numbers among the smallest parts in the partitions of 2n into two parts (see example). - _Wesley Ivan Hurt_, Jul 25 2014
%C A110660 For n > 0, a(n-1) is the sum of the smallest parts of the partitions of 2n into two distinct even parts. - _Wesley Ivan Hurt_, Dec 06 2017
%H A110660 G. C. Greubel, <a href="/A110660/b110660.txt">Table of n, a(n) for n = 0..5000</a>
%H A110660 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PronicNumber.html">Pronic Number</a>
%H A110660 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A110660 a(n) = floor(n/2) * (floor(n/2)+1).
%F A110660 a(n) = A028242(n) * A110654(n).
%F A110660 a(n) = A008805(n-2)*2, with A008805(-2) = A008805(-1) = 0.
%F A110660 From _Wesley Ivan Hurt_, Jul 25 2014: (Start)
%F A110660 G.f.: 2*x^2/((1-x)^3*(1+x)^2);
%F A110660 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), for n > 4;
%F A110660 a(n) = (2*n^2 + 2*n - 1 + (2*n + 1)*(-1)^n)/8. (End)
%F A110660 a(n) = Sum_{i=1..n; i even} i. - _Olivier Pirson_, Nov 05 2017
%e A110660 a(4) = 6; The partitions of 2*4 = 8 into two parts are: (7,1), (6,2), (5,3), (4,4). The sum of the even numbers from the smallest parts of these partitions gives: 2 + 4 = 6.
%e A110660 a(5) = 6; The partitions of 2*5 = 10 into two parts are: (9,1), (8,2), (7,3), (6,4), (5,5). The sum of the even numbers from the smallest parts of these partitions gives: 2 + 4 = 6.
%p A110660 A110660:=n->floor(n/2)*(floor(n/2)+1): seq(A110660(n), n=0..50); # _Wesley Ivan Hurt_, Jul 25 2014
%t A110660 Table[Floor[n/2] (Floor[n/2] + 1), {n, 0, 50}] (* _Wesley Ivan Hurt_, Jul 25 2014 *)
%t A110660 CoefficientList[Series[2*x^2/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jul 25 2014 *)
%t A110660 LinearRecurrence[{1,2,-2,-1,1},{0,0,2,2,6},60] (* _Harvey P. Dale_, Jan 23 2021 *)
%o A110660 (Magma) k:=1; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1,1], n in [1..30]]; // _Bruno Berselli_, Nov 14 2012
%o A110660 (PARI) a(n)=n\=2;n*(n+1) \\ _Charles R Greathouse IV_, Jul 05 2013
%Y A110660 Cf. A109613.
%Y A110660 Partial sums give A006584.
%K A110660 nonn,easy
%O A110660 0,3
%A A110660 _Reinhard Zumkeller_, Aug 05 2005
%E A110660 Typo in description (Name) fixed by _Harvey P. Dale_, Jul 09 2021