A229655 Quintisection a(5n+k) gives k-th differences of a for k=0..4 with a(n)=0 for n<4 and a(4)=1.

0, 0, 0, 0, 1, 0, 0, 0, 1, -4, 0, 0, 1, -3, 6, 0, 1, -2, 3, -4, 1, -1, 1, -1, 2, 0, 0, 0, 1, -8, 0, 0, 1, -7, 22, 0, 1, -6, 15, -28, 1, -5, 9, -13, 18, -4, 4, -4, 5, -11, 0, 0, 1, -6, 24, 0, 1, -5, 18, -46, 1, -4, 13, -28, 50, -3, 9, -15, 22, -33, 6, -6, 7

Offset: 0

Author

Alois P. Heinz, Sep 27 2013

Keywords

• sign
• look
• eigen

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..15625

Crossrefs

Cf. A005590, A229653, A229654, A229656, A229657, A229658, A229659, A229660.

Programs

Maple

a:= proc(n) option remember; local m, q;
      m:= irem(n, 5, 'q'); `if`(n<5, `if`(n=4, 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, 5]; If[n < 5, If[n == 4, 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, from Maple *)

Formula

a(5*n) = a(n),

a(5*n+1) = a(n+1) - a(n),

a(5*n+2) = a(n+2) - 2*a(n+1) + a(n),

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

a(5*n+4) = a(n+4) - 4*a(n+3) + 6*a(n+2) - 4*a(n+1) + a(n).