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 A291219 #19 Sep 08 2022 08:46:19
%S A291219 1,1,3,5,11,21,42,83,163,323,635,1255,2473,4880,9625,18985,37451,
%T A291219 73869,145715,287421,566954,1118331,2205947,4351307,8583091,16930447,
%U A291219 33395857,65874464,129939569,256310161,505580371,997274197,1967156763,3880282533,7653987242
%N A291219 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^3.
%C A291219 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 A291219 In the following guide to p-INVERT sequences using s = (1,0,1,0,1,...) = A000035, in some cases t(1,0,1,0,1,...) is a shifted version of the indicated sequence.
%C A291219 p(S) t(1,0,1,0,1,...)
%C A291219 1 - S A000045 (Fibonacci numbers)
%C A291219 1 - S^2 A147600
%C A291219 1 - S^3 A291217
%C A291219 1 - S^5 A291218
%C A291219 1 - S - S^2 A289846
%C A291219 1 - S - S^3 A291219
%C A291219 1 - S - S^4 A291220
%C A291219 1 - S^3- S^6 A291221
%C A291219 1 - S^2- S^3 A291222
%C A291219 1 - S^3- S^4 A291223
%C A291219 1 - 2S A052542
%C A291219 1 - 3S A006190
%C A291219 (1 - S)^2 A239342
%C A291219 (1 - S)^3 A276129
%C A291219 (1 - S)^4 A291224
%C A291219 (1 - S)^5 A291225
%C A291219 (1 - S)^6 A291226
%C A291219 1 - S - 2 S^2 A291227
%C A291219 1 - 2 S - 2 S^2 A291228
%C A291219 1 - 3 S - 2 S^2 A060801
%C A291219 (1 - S)(1 - 2 S) A291229
%C A291219 (1 - S)(1 - 2 S)(1 - 3 S) A291230
%C A291219 (1 - S)(1 - 2 S)(1 - 3 S)( 1 - 4 S) A291231
%C A291219 (1 - 2 S)^2 A291264
%C A291219 (1 - 3 S)^2 A291232
%C A291219 1 - S - S^2 - S^3 A291233
%C A291219 1 - S - S^2 - S^3 - S^4 A291234
%C A291219 1 - S - S^2 - S^3 - S^4 - S^5 A291235
%C A291219 (1 - S)(1 - 3 S) A291236
%C A291219 (1 - S)(1 - 2S)( 1 - 4S) A291237
%C A291219 (1 - S)^2 (1 - 2S) A291238
%C A291219 (1 - S^2) (1 - 2S) A291239
%C A291219 (1 - S^3)^2 A291240
%C A291219 1 - S - S^2 + S^3 A291241
%C A291219 1 - 2 S - S^2 + S^3 A291242
%C A291219 1 - 3 S + S^2 A291243
%C A291219 1 - 4 S + S^2 A291244
%C A291219 1 - 5 S + S^2 A291245
%C A291219 1 - 6 S + S^2 A291246
%C A291219 1 - S - S^2 - S^3 + S^4 A291247
%C A291219 1 - S - S^2 - S^3 - S^4 + S^5 A291248
%C A291219 1 - S - S^2 - S^3 + S^4 + S^5 A291249
%C A291219 1 - S - 2 S^2 + 2 S^3 A291250
%C A291219 1 - 3 S^2 + 2 S^3 A291251 (includes negative terms)
%C A291219 (1 - S^3)^3 A291252
%C A291219 (1 - S - S^2)^2 A291253
%C A291219 (1 - 2 S - S^2)^2 A291254
%C A291219 (1 - S - 2 S^2)^2 A291255
%H A291219 Clark Kimberling, <a href="/A291219/b291219.txt">Table of n, a(n) for n = 0..1000</a>
%H A291219 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-3,1,1)
%F A291219 G.f.: -(1 - x^2 + x^4)/(-1 + x + 3*x^2 - x^3 - 3*x^4 + x^5 + x^6).
%F A291219 a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 3*a(n-4) + a(n-5) + a(n-6) for n >= 7.
%t A291219 z = 60; s = x/(1 - x^2); p = 1 - s - s^3;
%t A291219 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
%t A291219 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291219 *)
%t A291219 LinearRecurrence[{1, 3, -1, -3, 1, 1}, {1, 1, 3, 5, 11, 21}, 50] (* _Vincenzo Librandi_, Aug 25 2017 *)
%o A291219 (Magma) I:=[1,1,3,5,11,21]; [n le 6 select I[n] else Self(n-1)+3*Self(n-2)-Self(n-3)-3*Self(n-4)+Self(n-5)+Self(n-6): n in [1..45]]; // _Vincenzo Librandi_, Aug 25 2017
%Y A291219 Cf. A000035, A290890, A291000.
%K A291219 nonn,easy
%O A291219 0,3
%A A291219 _Clark Kimberling_, Aug 24 2017