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%I #58 Sep 08 2022 08:46:04
%S 1,2,4,10,26,66,160,372,840,1864,4096,8944,19424,41952,90112,192576,
%T 409728,868480,1835008,3866368,8125952,17038848,35651584,74449920,
%U 155191296,322963456,671088640,1392504832,2885672960,5972680704,12348030976,25501384704
%N Binomial transform of A215495(n).
%C Companion to A218009.
%C Like any other sequence with a linear recurrence with constant coefficients, this sequence is periodic if read modulo some constant m. These Pisano period lengths for m>=1 are 1, 1, 8, 1, 20, 8, 168, 1, 24, 20, 440, 8, 156, 168, 40, 1, 272, 24, 1368, 20, ... [Curtz's comment reformulated and extended by _R. J. Mathar_, Oct 23 2012]
%C Let b(n) = a(n+1)-2*a(n), then b(n+3)-2*b(n+2) = A009545(n+2). - edited by _Michel Marcus_, Apr 24 2018
%H Vincenzo Librandi, <a href="/A217988/b217988.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 (6,-14,16,-8).
%F a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4) with n > 5.
%F a(n) = A218009(n) + A146559(n).
%F G.f.: (1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2)). - _Bruno Berselli_, Oct 22 2012
%F a(n) = 2^(n-3)*(3*n+2)+((1+i)^n+(1-i)^n)/4, where i=sqrt(-1) and n>1, with a(0)=1, a(1)=2.
%e a(n) and successive differences:
%e 1, 2, 4, 10, 26, 66, 160, 372, 840, 1864, 4096, ...
%e 1, 2, 6, 16, 40, 94, 212, 468, 1024, ...
%e 1, 4, 10, 24, 54, 118, 256, ...
%e 3, 6, 14, 30, 64, ...
%e 3, 8, 16, ...
%e 5, 8, ...
%e 3, ...
%t a[n_] := Sum[ Binomial[n, k]*If[ OddQ[k], k, k/2 + Boole[ Mod[k, 4] == 0]], {k, 0, n}]; Table[ a[n], {n, 0, 31}] (* _Jean-François Alcover_, Oct 17 2012 *)
%t CoefficientList[Series[(1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2 * (1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 15 2012 *)
%t LinearRecurrence[{6,-14,16,-8},{1,2,4,10,26,66},40] (* _Harvey P. Dale_, Aug 14 2018 *)
%o (Magma) I:=[1, 2, 4, 10, 26, 66]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 15 2012
%o (PARI) x='x+O('x^30); Vec((1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2))) \\ _G. C. Greubel_, Apr 23 2018
%Y Cf. A214282, A214283.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Oct 17 2012
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