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 A147600 #57 Mar 30 2024 23:07:41
%S A147600 1,0,3,0,8,0,21,0,55,0,144,0,377,0,987,0,2584,0,6765,0,17711,0,46368,
%T A147600 0,121393,0,317811,0,832040,0,2178309,0,5702887,0,14930352,0,39088169,
%U A147600 0,102334155,0,267914296,0,701408733,0,1836311903,0,4807526976,0
%N A147600 Expansion of 1/(1 - 3*x^2 + x^4).
%C A147600 S(n,sqrt(5)), with the Chebyshev polynomials A049310, is an integer sequence in the real quadratic number field Q(sqrt(5)) with basis numbers <1,phi>, phi:=(1+sqrt(5))/2. S(n,sqrt(5)) = A(n) + 2*B(n)*phi, with A(n) = A005013(n+1)*(-1)^n and B(n) = a(n-1), n>=0, with a(-1)=0. - _Wolfdieter Lang_, Nov 24 2010
%C A147600 The sequence (s(n)) given by s(0) = 0 and s(n) = a(n-1) for n > 0 is the p-INVERT of (0, 1, 0, 1, 0, 1, ...) using p(S) = 1 - S^2; see A291219. - _Clark Kimberling_, Aug 30 2017
%C A147600 From _Jean-François Alcover_, Sep 24 2017: (Start)
%C A147600 Consider this array of successive differences:
%C A147600 0, 0, 0, 1, 0, 3, 0, 8, 0, 21, ...
%C A147600 0, 0, 1, -1, 3, -3, 8, -8, 21, -21, ...
%C A147600 0, 1, -2, 4, -6, 11, -16, 29, -42, 76, ...
%C A147600 1, -3, 6, -10, 17, -27, 45, -71, 118, -186, ...
%C A147600 -4, 9, -16, 27, -44, 72, -116, 189, -304, 495, ...
%C A147600 13, -25, 43, -71, 116, -188, 305, -493, 799, -1291, ...
%C A147600 -38, 68, -114, 187, -304, 493, -798, 1292, -2090, 3383, ...
%C A147600 ...
%C A147600 First row = even-index Fibonacci numbers with interleaved zeros = this sequence right-shifted 3 positions.
%C A147600 Main diagonal = 0,0,-2,-10,-44,-188,-798,... = -A099919 right-shifted.
%C A147600 First upper subdiagonal = 0,1,4,17,72,305,1292,... = A001076 right-shifted.
%C A147600 Second upper subdiagonal = 0,-1,-6,-27,-116,-493,-2090,... = -A049651.
%C A147600 Third upper subdiagonal = 1,3,11,45,189,799,3383,... = A292278.
%C A147600 (End) (Comment based on an e-mail from _Paul Curtz_)
%H A147600 Colin Barker, <a href="/A147600/b147600.txt">Table of n, a(n) for n = 0..1000</a>
%H A147600 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A147600 O.g.f.: 1/(1 - 3*x^2 + x^4).
%F A147600 a(2*k) = F(2*(k+1)), a(2*k+1) = 0, k>=0, with F(n)=A000045(n). - _Richard Choulet_, Nov 13 2008
%F A147600 a(n) + a(n-1) + a(n-2) = A005013(n + 1). - _Michael Somos_, Apr 13 2012
%F A147600 a(n) = (2^(-2-n)*((1 + (-1)^n)*((-3+sqrt(5))*(-1+sqrt(5))^n + (1+sqrt(5))^n*(3+sqrt(5)))))/sqrt(5). - _Colin Barker_, Mar 28 2016
%e A147600 G.f. = 1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 + 144*x^10 + 377*x^12 + 987*x^14 + ...
%t A147600 f[x_]= -1 -x +x^2; CoefficientList[Series[-1/(x^2*f[x]*f[1/x]), {x,0,60}], x]
%t A147600 (* or *)
%t A147600 M={{0,1,0,0}, {0,0,1,0}, {0,0,0,1}, {-1,0,3,0}}; v[0]= {1,0,3,0}; v[n_]:= v[n]= M.v[n-1]; Table[v[n][[1]], {n,0,60}]
%t A147600 LinearRecurrence[{0,3,0,-1}, {1,0,3,0}, 60] (* _Jean-François Alcover_, Sep 23 2017 *)
%o A147600 (PARI) Vec(1/(1 - 3*x^2 + x^4)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A147600 (Magma) [(1+(-1)^n)*Fibonacci(n+2)/2: n in [0..60]]; // _G. C. Greubel_, Oct 25 2022
%o A147600 (SageMath) [((n+1)%2)*fibonacci(n+2) for n in range(60)] # _G. C. Greubel_, Oct 25 2022
%Y A147600 Cf. A000045, A001076, A001906, A005013, A049310.
%Y A147600 Cf. A049651, A088305, A099919, A291219, A292278.
%K A147600 nonn,easy
%O A147600 0,3
%A A147600 _Roger L. Bagula_, Nov 08 2008