%I #35 Dec 05 2019 08:22:09
%S 1,0,1,0,2,3,8,14,29,56,114,227,456,910,1821,3640,7282,14563,29128,
%T 58254,116509,233016,466034,932067,1864136,3728270,7456541,14913080,
%U 29826162,59652323,119304648,238609294,477218589,954437176,1908874354,3817748707
%N a(n+3) = 2^n - a(n), a(0)=a(2)=1, a(1)=0 for n >= 0.
%C The array of a(n) and its repeated differences:
%C 1, 0, 1, 0, 2, 3, 8, 14, ...
%C -1, 1, -1, 2, 1, 5, 6, 15, ...
%C 2, -2, 3, -1, 4, 1, 9, 12, ...
%C -4, 5, -4, 5, -3, 8, 3, 19, ...
%C 9, -9, 9, -8, 11, -5, 16, 5, ...
%C -18, 18, -17, 19, -16, 21, -11, 32, ...
%C 36, -35, 36, -35, 37, -32, 43, -21, ...
%C -71, 71, -71, 72, -69, 75, -64, 85, ...
%C ...
%C The recurrence is the same for every row.
%C From _Jean-François Alcover_, Nov 28 2019: (Start)
%C It appears that, when odd, a(n) is never a multiple of 5.
%C Main and 3rd upper diagonals of the difference array are A001045 (Jacobsthal numbers); first upper diagonal is negated A001045; second upper diagonal is A000079 (powers of 2); 4th upper diagonal is A062092.
%C (End)
%H Colin Barker, <a href="/A328881/b328881.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).
%F a(n+1) - 2*a(n) = period 6: repeat [-2, 1, -2, 2, -1, 2].
%F a(n+12) - a(n) = 455*2^n.
%F From _Colin Barker_, Oct 29 2019: (Start)
%F G.f.: (1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)).
%F a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
%F (End)
%F a(n+2) - a(n) = A024495(n).
%F a(n+6) - a(n) = 7*2^n.
%F a(n+9) + a(n) = 57*2^n.
%F a(n) = A113405(n) + A092220(n+5).
%F 9*a(n) = 2^n + 5*(-1)^n + 3*A010892(n). - _R. J. Mathar_, Nov 28 2019
%t a[0] = a[2] = 1; a[1] = 0; a[n_] := a[n] = 2^(n - 3) - a[n - 3]; Array[a, 36, 0] (* _Amiram Eldar_, Nov 06 2019 *)
%o (PARI) Vec((1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ _Colin Barker_, Oct 29 2019
%Y Cf. A024495, A062092, A131714.
%Y Cf. A015565, A092220, A113405.
%K nonn,easy
%O 0,5
%A _Paul Curtz_, Oct 29 2019