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Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).
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%I #33 Sep 08 2022 08:46:04

%S 0,1,3,7,15,31,62,124,248,496,992,1985,3971,7943,15887,31775,63550,

%T 127100,254200,508400,1016800,2033601,4067203,8134407,16268815,

%U 32537631,65075262,130150524,260301048,520602096,1041204192,2082408385,4164816771,8329633543

%N Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).

%C This is the case k=5 of a(n) + a(n+k) = 2^(n+k) - 1 = A000225(n+k). The sequences A000975, A077854, A153234 and A224520 correspond to cases k=1,2,3 and 4, respectively.

%H G. C. Greubel, <a href="/A224521/b224521.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,-1,3,-2).

%F a(n) + a(n+5) = 2^(n+5) - 1.

%F From _Joerg Arndt_, Apr 09 2013: (Start)

%F G.f.: x/((1-x)*(1+x)*(1-2*x)*(1-x+x^2-x^3+x^4)).

%F a(n) = +3*a(n-1) -2*a(n-2) -1*a(n-5) +3*a(n-6) -2*a(n-7). (End)

%F a(n) = floor(2^(n+5)/33). - _Karl V. Keller, Jr._, Jul 03 2021

%t CoefficientList[Series[x/((1-x)*(1-2*x)*(1+x^5)), {x,0,40}], x] (* _G. C. Greubel_, Oct 11 2017 *)

%t LinearRecurrence[{3,-2,0,0,-1,3,-2},{0,1,3,7,15,31,62},40] (* _Harvey P. Dale_, Apr 29 2020 *)

%o (PARI) my(x='x+O('x^40)); concat([0], Vec(x/((1-x)*(1-2*x)*(1+x^5)))) \\ _G. C. Greubel_, Oct 11 2017

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*(1-2*x)*(1+x^5)) )); // _G. C. Greubel_, Jun 06 2019

%o (Sage) (x/((1-x)*(1-2*x)*(1+x^5))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 06 2019

%o (Python) print([2**(n+5)//33 for n in range(31)]) # _Karl V. Keller, Jr._, Jul 03 2021

%Y Cf. A000975, A077854, A153234, A224520.

%K nonn,easy

%O 0,3

%A _Arie Bos_, Apr 09 2013