A229653 Trisection a(3n+k) gives k-th differences of a for k=0..2 with a(n)=0 for n<2 and a(2)=1.

0, 0, 1, 0, 1, -2, 1, -1, 2, 0, 1, -4, 1, -3, 6, -2, 3, -5, 1, -2, 5, -1, 3, -5, 2, -2, 3, 0, 1, -6, 1, -5, 10, -4, 5, -9, 1, -4, 13, -3, 9, -17, 6, -8, 13, -2, 5, -13, 3, -8, 14, -5, 6, -9, 1, -3, 10, -2, 7, -13, 5, -6, 10, -1, 4, -12, 3, -8, 15, -5, 7, -11

Offset: 0

Author

Alois P. Heinz, Sep 27 2013

Keywords

• sign
• eigen
• look

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..19683

Crossrefs

Cf. A005590, A229654, A229655, A229656, A229657, A229658, A229659, A229660.

Programs

Maple

a:= proc(n) option remember; local m, q;
      m:= irem(n, 3, 'q'); `if`(n<3, `if`(n=2, 1, 0),
      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}, {q, m} = QuotientRemainder[n, 3]; If[n < 3, If[n == 2, 1, 0], Sum[a[q + m - j]*(-1)^j*Binomial[m, j], {j, 0, m}]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, translated from Maple *)

Formula

a(3*n) = a(n),

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

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