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%I #21 Jan 15 2024 22:11:37
%S 0,0,2,0,4,4,10,6,14,14,24,18,30,30,44,36,52,52,70,60,80,80,102,90,
%T 114,114,140,126,154,154,184,168,200,200,234,216,252,252,290,270,310,
%U 310,352,330,374,374,420,396,444,444,494,468,520,520,574,546,602,602
%N Sum of the larger parts in the partitions of n into two distinct parts with the larger part even.
%H Robert Israel, <a href="/A296805/b296805.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F a(n) = Sum_{i=1..floor((n-1)/2)} (n-i) * ((n-i+1) mod 2).
%F G.f.: 2*x^3*(1-x+2*x^2+x^4)/((1-x)*(1-x^4)^2). - _Robert Israel_, Dec 20 2017
%F From _Wesley Ivan Hurt_, Jan 15 2024: (Start)
%F a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).
%F a(n) = (1-10*n+6*n^2+(3-10*n)*(-1)^n-2*(2*n+1+(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/32. (End)
%e a(12) = 18; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). The sum of the even numbers among the larger parts gives 10 + 8 = 18.
%p f:= gfun:-rectoproc({a(n-9)-a(n-8)-2*a(n-5)+2*a(n-4)+a(n-1)-a(n)=0,seq(a(n)=[0,0,2,0,4,4,10,6,14][n],n=1..9)},a(n),remember):
%p map(f, [$1..100]); # _Robert Israel_, Dec 21 2017
%t Table[Sum[(n - i) Mod[n - i + 1, 2], {i, Floor[(n - 1)/2]}], {n, 80}]
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Dec 20 2017
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