A229819 Even bisection gives sequence a itself, n->a(2*(4*n+k)-1) gives k-th differences of a for k=1..4 with a(n)=n for n<2.

0, 1, 1, -1, 1, -1, -1, 7, 1, 0, -1, -2, -1, 6, 7, -14, 1, -2, 0, 4, -1, -8, -2, 14, -1, 2, 6, -4, 7, 6, -14, 0, 1, -2, -2, 2, 0, 6, 4, -28, -1, 0, -8, 8, -2, -22, 14, 41, -1, 8, 2, -14, 6, 19, -4, -24, 7, -6, 6, 5, -14, -5, 0, 5, 1, -1, -2, 0, -2, 0, 2, 2, 0

Offset: 0

Author

Alois P. Heinz, Sep 30 2013

Keywords

• sign
• eigen

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A005590, A229817, A229818, A229820, A229821, A229822, A229823, A229824, A229825.

Programs

Maple

a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 8, 'q')+1)/2;
      `if`(n<2, n, `if`(irem(n, 2, 'r')=0, a(r),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m)))
    end:
seq(a(n), n=0..100);

Mathematica

a[n_] := a[n] = Module[{m, q, r, q2, r2}, {q, r} = QuotientRemainder[n, 8]; m = (r+1)/2; If[n<2, n, {q2, r2} = QuotientRemainder[n, 2]; If[r2 == 0, a[q2], Sum[a[q+m-j]*(-1)^j*Binomial[m, j], {j, 0, m}]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017, translated from Maple *)

Formula

a(2*n) = a(n),

a(8*n+1) = a(n+1) - a(n),

a(8*n+3) = a(n+2) - 2*a(n+1) + a(n),

a(8*n+5) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n).

a(8*n+7) = a(n+4) - 4*a(n+3) + 6*a(n+2) - 4*a(n+1) + a(n).