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A224521 Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).

%I A224521 #33 Sep 08 2022 08:46:04
%S A224521 0,1,3,7,15,31,62,124,248,496,992,1985,3971,7943,15887,31775,63550,
%T A224521 127100,254200,508400,1016800,2033601,4067203,8134407,16268815,
%U A224521 32537631,65075262,130150524,260301048,520602096,1041204192,2082408385,4164816771,8329633543
%N A224521 Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).
%C A224521 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 A224521 G. C. Greubel, <a href="/A224521/b224521.txt">Table of n, a(n) for n = 0..1000</a>
%H A224521 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,-1,3,-2).
%F A224521 a(n) + a(n+5) = 2^(n+5) - 1.
%F A224521 From _Joerg Arndt_, Apr 09 2013: (Start)
%F A224521 G.f.: x/((1-x)*(1+x)*(1-2*x)*(1-x+x^2-x^3+x^4)).
%F A224521 a(n) = +3*a(n-1) -2*a(n-2) -1*a(n-5) +3*a(n-6) -2*a(n-7). (End)
%F A224521 a(n) = floor(2^(n+5)/33). - _Karl V. Keller, Jr._, Jul 03 2021
%t A224521 CoefficientList[Series[x/((1-x)*(1-2*x)*(1+x^5)), {x,0,40}], x] (* _G. C. Greubel_, Oct 11 2017 *)
%t A224521 LinearRecurrence[{3,-2,0,0,-1,3,-2},{0,1,3,7,15,31,62},40] (* _Harvey P. Dale_, Apr 29 2020 *)
%o A224521 (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 A224521 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*(1-2*x)*(1+x^5)) )); // _G. C. Greubel_, Jun 06 2019
%o A224521 (Sage) (x/((1-x)*(1-2*x)*(1+x^5))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 06 2019
%o A224521 (Python) print([2**(n+5)//33 for n in range(31)]) # _Karl V. Keller, Jr._, Jul 03 2021
%Y A224521 Cf. A000975, A077854, A153234, A224520.
%K A224521 nonn,easy
%O A224521 0,3
%A A224521 _Arie Bos_, Apr 09 2013