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%I #24 Aug 23 2016 21:27:42
%S 0,0,1,3,7,17,43,108,273,708,1867,4955,13256,35790,97340,266240,
%T 732014,2022558,5612579,15634288,43702232,122550885,344661924,
%U 971908613,2747404212,7784038617,22100387619,62869809733,179173559128,511497066733,1462522478549,4188024794407
%N Number of different equations that can be made by summing numbers from 1 to n and using every number not more than once.
%C The summands of each side are in increasing order and the minimum of all summands is on the left side.
%H Alois P. Heinz, <a href="/A161943/b161943.txt">Table of n, a(n) for n = 1..940</a>
%F a(n) ~ 3^(n+1) / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 11 2014
%e a(3) = 1, as the only equation we can make by summing numbers from the set {1, 2, 3} is 1+2=3. a(4) = 3, as we can make three equations: 1+2=3, 1+3=4, 1+4=2+3.
%p b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
%p if n>m then 0
%p elif n=m then 1
%p else b(n, i-1) +b(abs(n-i), i-1) +b(n+i, i-1)
%p fi
%p end:
%p a:= proc(n) option remember;
%p `if`(n>2, b(n, n-1)+ a(n-1), 0)
%p end:
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Aug 31 2009, revised Sep 16 2011
%t Table[(Length[ Select[Range[0, 3^n - 1], Apply[Plus, Pick[Range[n], PadLeft[IntegerDigits[ #, 3], n], 1]] == Apply[Plus, Pick[Range[n], PadLeft[IntegerDigits[ #, 3], n], 2]] &]] - 1)/ 2, {n, 14}]
%Y Column k=2 of A196231.
%K nonn
%O 1,4
%A _Tanya Khovanova_, Jun 22 2009
%E More terms from _Alois P. Heinz_, Aug 31 2009
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