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 A133037 #31 Sep 30 2024 15:06:28
%S A133037 1,0,0,1,0,1,1,1,4,4,9,16,25,49,81,144,256,441,784,1369,2401,4225,
%T A133037 7396,12996,22801,40000,70225,123201,216225,379456,665856,1168561,
%U A133037 2050624,3598609,6315169,11082241,19448100,34128964,59892121,105103504,184443561,323676081
%N A133037 a(n) = A000931(n)^2.
%C A133037 a(n+3) is the number of tilings of an n-board (a board with dimensions n X 1) with (1/2,1/2;2)-combs and (1/2,1/2;3)-combs. A (w,g;m)-comb is a tile composed of m pieces of dimensions w X 1 separated horizontally by gaps of width g. - _Michael A. Allen_, Sep 25 2024
%H A133037 Michael A. Allen and Kenneth Edwards, <a href="https://doi.org/10.1080/03081087.2022.2107979">Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices</a>, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
%H A133037 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,1,-1).
%F A133037 a(n) = A000931(n)^2.
%F A133037 a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6).
%F A133037 G.f.: (x^5+x^2+x-1)/(-x^6+x^5-x^4+x^3+x^2+x-1).
%F A133037 a(n) = a(n-2) + a(n-3) + 2*Sum_{r=8..n} ( A000930(r-8)*a(n+3-r) ) for n >= 3. - _Michael A. Allen_, Sep 25 2024
%e A133037 a(10)=9 because Padovan(10)=3 and 3^2=9.
%t A133037 a[0] = a[3] = a[5] = a[6] = 1; a[1] = a[2] = a[4] = 0; a[n_Integer] := a[n] = 2*a[n - 2] + 2*a[n - 3] - a[n - 7]; Table[a[i], {i, 0, 40}] (* _Olivier Gérard_, Jul 05 2011 *)
%t A133037 Table[RootSum[-1 - # + #^3 &, #^n (5 - 6 # + 4 #^2) &]^2/529, {n, 0,
%t A133037 40}] (* _Eric W. Weisstein_, Apr 16 2018 *)
%t A133037 LinearRecurrence[{1, 1, 1, -1, 1, -1}, {1, 0, 0, 1, 0, 1}, 40] (* _Eric W. Weisstein_, Apr 16 2018 *)
%o A133037 (PARI) Vec(O(x^20)+(1-x-x^2-x^5)/(1-x-x^2-x^3+x^4-x^5+x^6)) \\ _Charles R Greathouse IV_, Jul 05 2011
%Y A133037 Cf. A000290, A001248, A007598. Padovan sequence: A000931.
%K A133037 easy,nonn
%O A133037 0,9
%A A133037 _Omar E. Pol_, Nov 02 2007