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%I #51 Jun 24 2022 04:34:48
%S 0,1,1,2,4,9,17,38,72,161,305,682,1292,2889,5473,12238,23184,51841,
%T 98209,219602,416020,930249,1762289,3940598,7465176,16692641,31622993,
%U 70711162,133957148,299537289,567451585,1268860318,2403763488,5374978561
%N Expansion of x*(1 + x - 2*x^2) / ( 1 - 4*x^2 - x^4).
%C Based on fact that cube root of (2 +- 1 sqrt(5)) = sixth root of (9 +- 4 sqrt(5)) = ninth root of (38 +- 17 sqrt(5)) = ... = phi or 1/phi, where phi is the golden ratio.
%C Osler gives the first three of the above equalities with phi on page 27, stating they are simplified expressions from Ramanujan, but without hinting that the series continues.
%C Bisections: A001076 and A001077.
%H G. C. Greubel, <a href="/A059973/b059973.txt">Table of n, a(n) for n = 0..1000</a>
%H T. J. Osler, <a href="http://www.jstor.org/stable/2691150">Cardan polynomials and the reduction of radicals</a>, Math. Mag., 74 (No. 1, 2001), 26-32.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,1).
%F From _Michael Somos_, Aug 11 2009: (Start)
%F a(2*n) = A001076(n).
%F a(2*n+1) = A001077(n). (End)
%F Recurrence: a(n) = 4*a(n-2) + a(n-4) for n >= 4; a(0)=0, a(1)=a(2)=1, a(3)=2. - _Werner Schulte_, Oct 03 2015
%F From _Altug Alkan_, Oct 06 2015: (Start)
%F a(2n) = Sum_{k=0..2n-1} a(k).
%F a(2n+1) = A001076(n-1) + Sum_{k=0..2n} a(k), n>0. (End)
%e G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 17*x^6 + 38*x^7 + 72*x^8 + 161*x^9 + ... - _Michael Somos_, Aug 11 2009
%t CoefficientList[ Series[(x +x^2 -2x^3)/(1 -4x^2 -x^4), {x, 0, 33}], x]
%t LinearRecurrence[{0,4,0,1}, {0,1,1,2}, 50] (* _Vincenzo Librandi_, Oct 10 2015 *)
%o (PARI) {a(n) = if( n<0, n = -n; polcoeff( (-2*x + x^2 + x^3) / (1 + 4*x^2 - x^4) + x*O(x^n), n), polcoeff( (x + x^2 - 2*x^3) / ( 1 - 4*x^2 - x^4) + x*O(x^n), n))} /* _Michael Somos_, Aug 11 2009 */
%o (PARI) a(n) = if (n < 4, fibonacci(n), 4*a(n-2) + a(n-4));
%o vector(50, n, a(n-1)) \\ _Altug Alkan_, Oct 04 2015
%o (Magma) I:=[0,1,1,2]; [n le 4 select I[n] else 4*Self(n-2)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Oct 10 2015
%o (Sage)
%o def a(n): return fibonacci(n) if (n<4) else 4*a(n-2) + a(n-4)
%o [a(n) for n in [0..40]] # _G. C. Greubel_, Jul 12 2021
%Y Cf. A000045, A001076, A001077.
%Y Cf. A179319, A183555, A183556.
%K easy,nonn
%O 0,4
%A H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001
%E Edited by _Randall L Rathbun_, Jan 11 2002
%E More terms from _Sascha Kurz_, Jan 31 2003
%E I made the old definition into a comment and gave the g.f. as an explicit definition. - _N. J. A. Sloane_, Jan 05 2011
%E Moved g.f. from _Michael Somos_, into name to match terms. - _Paul D. Hanna_, Jan 12 2011
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