A261062 Number of solutions to c(1)*prime(2) + ... + c(2n-1)*prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

0, 0, 1, 0, 6, 8, 30, 121, 385, 1102, 4207, 13263, 48904, 164298, 610450, 2108897, 7592564, 27444148, 100851443, 365507140, 1344593522, 4960584613, 18435632285, 68320148701, 254166868115, 951593812462, 3568369245595, 13386056545363, 50416752718382

Offset: 1

Author

M. F. Hasler, Aug 08 2015

Keywords

• nonn

Comments

There cannot be a solution for an even number of terms on the l.h.s. because all terms are odd but the r.h.s. is odd, too.

Examples

			a(1) = a(2) = 0 because prime(2) and prime(2) +- prime(3) +- prime(4) are always different from -1.
a(3) = 1 because the solution prime(2) + prime(3) - prime(4) + prime(5) - prime(6) = -1 is the only one involving prime(2) through prime(6).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A261061, A261063 and A261044 (starting with prime(1), prime(3) and prime(4)), A022894, ..., A022904, A022920, A083309 (r.h.s. = 0, 1 or 2), A261057, A261059, A261060, A261045 (r.h.s. = -2).

Programs

Maple

s:= proc(n) option remember;
      `if`(n<3, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=2, 1,
      b(abs(n-ithprime(i)), i-1)+b(n+ithprime(i), i-1)))
    end:
a:= n-> b(4, 2*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

Mathematica

s[n_] := s[n] = If[n < 3, 0, Prime[n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 2, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]];
a[n_] := b[4, 2n];
Array[a, 30] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

PARI

A261062(n,rhs=-1,firstprime=2)={rhs-=prime(firstprime);my(p=vector(2*n-2+bittest(rhs,0),i,prime(i+firstprime)));sum(i=1,2^#p-1,sum(j=1,#p,(-1)^bittest(i,j-1)*p[j])==rhs)} \\ For illustrative purpose; too slow for n >> 10.

Formula

a(n) = [x^4] Product_{k=3..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

Extensions

a(14)-a(29) from Alois P. Heinz, Aug 08 2015