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A229818 Even bisection gives sequence a itself, n->a(2*(3*n+k)-1) gives k-th differences of a for k=1..3 with a(n)=n for n<2.

%I A229818 #25 Dec 12 2024 23:09:39
%S A229818 0,1,1,-1,1,-1,-1,0,1,-2,-1,6,-1,-2,0,4,1,-8,-2,2,-1,-4,6,6,-1,-2,-2,
%T A229818 2,0,-1,4,0,1,1,-8,-1,-2,1,2,0,-1,-4,-4,1,6,-4,6,8,-1,-3,-2,4,-2,2,2,
%U A229818 1,0,6,-1,-20,4,7,0,-14,1,20,1,-7,-8,6,-1,-3,-2,-1
%N A229818 Even bisection gives sequence a itself, n->a(2*(3*n+k)-1) gives k-th differences of a for k=1..3 with a(n)=n for n<2.
%H A229818 Alois P. Heinz, <a href="/A229818/b229818.txt">Table of n, a(n) for n = 0..10000</a>
%F A229818 a(2*n)   = a(n),
%F A229818 a(6*n+1) = a(n+1) - a(n),
%F A229818 a(6*n+3) = a(n+2) - 2*a(n+1) + a(n),
%F A229818 a(6*n+5) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n).
%p A229818 a:= proc(n) option remember; local m, q, r;
%p A229818       m:= (irem(n, 6, 'q')+1)/2;
%p A229818       `if`(n<2, n, `if`(irem(n, 2, 'r')=0, a(r),
%p A229818       add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m)))
%p A229818     end:
%p A229818 seq(a(n), n=0..100);
%t A229818 a[n_] := a[n] = Module[{m, q, r, q2, r2}, {q, r} = QuotientRemainder[n, 6]; 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 *)
%o A229818 (PARI) {M=Map(); a(n)= n&&n>>=valuation(n, 2); my(r); mapisdefined(M, n, &r) && return(r); r=if(n<2, n, my(m=n%6, k=n\6); if(1==m, a(k+1)-a(k), 3==m, a(k+2)-2*a(k+1)+a(k), a(k+3)-3*a(k+2)+3*a(k+1)-a(k))); mapput(~M, n, r); r;} \\ _Ruud H.G. van Tol_, Nov 19 2024
%Y A229818 Cf. A005590, A229817, A229819, A229820, A229821, A229822, A229823, A229824, A229825.
%K A229818 sign,look,hear,eigen
%O A229818 0,10
%A A229818 _Alois P. Heinz_, Sep 30 2013