A161943 Number of different equations that can be made by summing numbers from 1 to n and using every number not more than once.

0, 0, 1, 3, 7, 17, 43, 108, 273, 708, 1867, 4955, 13256, 35790, 97340, 266240, 732014, 2022558, 5612579, 15634288, 43702232, 122550885, 344661924, 971908613, 2747404212, 7784038617, 22100387619, 62869809733, 179173559128, 511497066733, 1462522478549, 4188024794407

Offset: 1

Author

Tanya Khovanova, Jun 22 2009

Keywords

• nonn

Comments

The summands of each side are in increasing order and the minimum of all summands is on the left side.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..940

Crossrefs

Column k=2 of A196231.

Programs

Maple

b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      if n>m then 0
    elif n=m then 1
    else b(n, i-1) +b(abs(n-i), i-1) +b(n+i, i-1)
      fi
    end:
a:= proc(n) option remember;
      `if`(n>2, b(n, n-1)+ a(n-1), 0)
    end:
seq(a(n), n=1..40); # Alois P. Heinz, Aug 31 2009, revised Sep 16 2011

Mathematica

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}]

Formula

a(n) ~ 3^(n+1) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 11 2014

Extensions

More terms from Alois P. Heinz, Aug 31 2009