This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A291012 #12 Jun 05 2023 07:21:53
%S A291012 2,7,22,68,208,632,1912,5768,17368,52232,156952,471368,1415128,
%T A291012 4247432,12746392,38247368,114758488,344308232,1032990232,3099101768,
%U A291012 9297567448,27893226632,83680728472,251044282568,753137042008,2259419514632,6778275321112
%N A291012 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^2)*(1 - 2*S).
%C A291012 Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
%C A291012 See A291000 for a guide to related sequences.
%H A291012 Clark Kimberling, <a href="/A291012/b291012.txt">Table of n, a(n) for n = 0..1000</a>
%H A291012 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F A291012 G.f.: (2 - 3 x - x^2)/(1 - 5*x + 6*x^2).
%F A291012 a(n) = 5*a(n-1) - 6*a(n-2) for n >= 4.
%F A291012 a(n) = (16*3^n - 3*2^n) / 6 for n > 0. - _Colin Barker_, Aug 23 2017
%F A291012 E.g.f.: (1/6)*(-1 - 3*exp(2*x) + 16*exp(3*x)). - _G. C. Greubel_, Jun 04 2023
%t A291012 z = 60; s = x/(1-x); p = (1-s)^2(1-2s);
%t A291012 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
%t A291012 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* this sequence *)
%t A291012 LinearRecurrence[{5,-6}, {2,7,22}, 40] (* _G. C. Greubel_, Jun 04 2023 *)
%o A291012 (PARI) Vec((2 -3*x -x^2)/((1-2*x)*(1-3*x)) + O(x^30)) \\ _Colin Barker_, Aug 23 2017
%o A291012 (Magma) [2] cat [8*3^(n-1) - 2^(n-1): n in [1..40]]; // _G. C. Greubel_, Jun 04 2023
%o A291012 (SageMath) [8*3^(n-1) - 2^(n-1) - int(n==0)/6 for n in range(41)] # _G. C. Greubel_, Jun 04 2023
%Y A291012 Cf. A000012, A033453, A289780, A291000.
%K A291012 nonn,easy
%O A291012 0,1
%A A291012 _Clark Kimberling_, Aug 23 2017