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A229654 Quadrisection a(4n+k) gives k-th differences of a for k=0..3 with a(n)=0 for n<3 and a(3)=1.

%I A229654 #20 Jun 16 2018 19:09:18
%S A229654 0,0,0,1,0,0,1,-3,0,1,-2,3,1,-1,1,0,0,0,1,-6,0,1,-5,12,1,-4,7,-9,-3,3,
%T A229654 -2,-2,0,1,-4,12,1,-3,8,-15,-2,5,-7,7,3,-2,0,4,1,-2,4,-7,-1,2,-3,4,1,
%U A229654 -1,1,-1,0,0,0,1,0,0,1,-9,0,1,-8,21,1,-7,13,-18
%N A229654 Quadrisection a(4n+k) gives k-th differences of a for k=0..3 with a(n)=0 for n<3 and a(3)=1.
%H A229654 Alois P. Heinz, <a href="/A229654/b229654.txt">Table of n, a(n) for n = 0..65536</a>
%F A229654 a(4*n)   = a(n),
%F A229654 a(4*n+1) = a(n+1) - a(n),
%F A229654 a(4*n+2) = a(n+2) - 2*a(n+1) + a(n),
%F A229654 a(4*n+3) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n).
%p A229654 a:= proc(n) option remember; (m-> `if`(n<4, `if`(n=3, 1, 0), add(
%p A229654        a(q+m-j)*(-1)^j*binomial(m, j), j=0..m)))(irem(n, 4, 'q'))
%p A229654     end:
%p A229654 seq(a(n), n=0..100);
%t A229654 a[n_] := a[n] = Module[{ m, q}, {q, m} = QuotientRemainder[n, 4]; If[n < 4, If[n == 3, 1, 0], Sum[a[q + m - j]*(-1)^j*Binomial[m, j], {j, 0, m}]]];
%t A229654 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 09 2018, from Maple *)
%Y A229654 Cf. A005590, A229653, A229655, A229656, A229657, A229658, A229659, A229660.
%K A229654 sign,eigen,look
%O A229654 0,8
%A A229654 _Alois P. Heinz_, Sep 27 2013