A397489 - OEIS

%I #16 Jun 29 2026 12:14:28

%S 1,1,0,0,1,1,0,0,1,3,0,0,12,12,0,0,65,115,0,0,507,972,0,0,4931,8586,0,

%T 0,49845,86086,0,0,517481,931253,0,0,5643431,10439967,0,0,64489578,

%U 120255030,0,0,762585871,1422072284,0,0,9238616655,17252409833,0,0,114139783947,214024316649

%N Number of ways to write n as a sum n = 0 + 1 + 2 +- 3 +- 4 ... +- n with nonnegative partial sums.

%H Alois P. Heinz, <a href="/A397489/b397489.txt">Table of n, a(n) for n = 0..1001</a>

%F a(n) = A397488(n,n).

%F a(n) = 0 <=> n mod 4 >= 2.

%e a(4) = 1: 0+1+2-3+4.

%e a(9) = 3: 0+1+2-3+4+5-6+7+8-9, 0+1+2-3+4+5+6-7-8+9, 0+1+2+3-4+5-6+7-8+9.

%e a(12) = 12: 0+1+2-3+4+5+6-7+8+9+10-11-12, 0+1+2-3+4+5+6+7-8+9-10+11-12, 0+1+2-3+4+5+6+7+8-9-10-11+12, 0+1+2+3-4+5-6+7+8+9+10-11-12, 0+1+2+3-4+5+6-7+8+9-10+11-12, 0+1+2+3-4+5+6+7-8-9+10+11-12, 0+1+2+3-4+5+6+7-8+9-10-11+12, 0+1+2+3+4-5+6-7+8-9+10+11-12, 0+1+2+3+4-5+6-7+8+9-10-11+12, 0+1+2+3+4-5+6+7-8-9+10-11+12, 0+1+2+3+4+5-6-7+8-9+10-11+12, 0+1+2+3+4+5+6+7+8+9-10-11-12.

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(i=0,

%p      `if`(n=0, 1, 0), b(n+i, i-1)+ `if`(i>n, 0, b(n-i, i-1))))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..53);

%Y Main diagonal of A397488.

%K nonn

%O 0,10

%A _Alois P. Heinz_, Jun 27 2026