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A289261 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=13/8.

%I A289261 #19 Nov 23 2024 05:44:31
%S A289261 1,3,5,9,17,30,52,91,160,281,494,871,1537,2711,4782,8437,14885,26258,
%T A289261 46320,81712,144145,254277,448555,791273,1395843,2462330,4343664,
%U A289261 7662429,13516885,23844416,42062667,74200520,130893196,230901729,407321472,718534172
%N A289261 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=13/8.
%C A289261 Conjecture: the sequence is strictly increasing.
%H A289261 Colin Barker, <a href="/A289261/b289261.txt">Table of n, a(n) for n = 0..1000</a>
%H A289261 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-2,1,-2,2).
%F A289261 G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = 13/8 and [ ] = floor.
%F A289261 G.f.: (1 - x)*(1 + x)^2*(1 + x^2)*(1 + x^4) / (1 - 2*x + x^2 - 2*x^3 + 2*x^4 - x^5 + 2*x^6 - 2*x^7). - _Colin Barker_, Jul 14 2017
%t A289261 r = 13/8;
%t A289261 u = 1000; (* # initial terms from given series *)
%t A289261 v = 100;   (* # coefficients in reciprocal series *)
%t A289261 CoefficientList[Series[1/Sum[Floor[r*(k + 1)] (-x)^k, {k, 0, u}], {x, 0, v}], x]
%t A289261 LinearRecurrence[{2,-1,2,-2,1,-2,2},{1,3,5,9,17,30,52,91,160,281},40] (* _Harvey P. Dale_, Aug 04 2024 *)
%o A289261 (PARI) Vec((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x^4) / (1 - 2*x + x^2 - 2*x^3 + 2*x^4 - x^5 + 2*x^6 - 2*x^7) + O(x^50)) \\ _Colin Barker_, Jul 20 2017
%Y A289261 Cf. A078140 (includes guide to related sequences), A289266.
%K A289261 nonn,easy
%O A289261 0,2
%A A289261 _Clark Kimberling_, Jul 14 2017