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A096748 Expansion of (1+x)^2/(1-x^2-x^4).

%I A096748 #43 Apr 20 2024 10:28:30
%S A096748 1,2,2,2,3,4,5,6,8,10,13,16,21,26,34,42,55,68,89,110,144,178,233,288,
%T A096748 377,466,610,754,987,1220,1597,1974,2584,3194,4181,5168,6765,8362,
%U A096748 10946,13530,17711,21892,28657,35422,46368,57314,75025,92736,121393,150050
%N A096748 Expansion of (1+x)^2/(1-x^2-x^4).
%C A096748 The ratio a(n+1) / a(n) increasingly approximates two constants connected to the golden ratio phi = (1 + sqrt(5))/2: (phi+1)/2 = 1.30901699... = A239798 and (phi-1)*2 = 1.23606797... = A134972, according to whether n is odd or even. - _Davide Rotondo_, Jul 31 2020
%H A096748 Harvey P. Dale, <a href="/A096748/b096748.txt">Table of n, a(n) for n = 0..1000</a>
%H A096748 Davide Rotondo, <a href="http://daviderotondo.altervista.org/alterpages/files/perfinoicapellisonotuttticontatidiDavideRotondo.pdf">Perfino I Capelli Sono Tutti Contati</a> (in Italian), see p. 11.
%H A096748 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).
%F A096748 a(n) = a(n-2) + a(n-4).
%F A096748 a(n) = 2*F((n+1)/2)*(1-(-1)^n)/2 + F((n+4)/2)*(1+(-1)^n)/2.
%F A096748 a(2*n) = A000045(n+2); a(2*n+1) = 2*A000045(n+1).
%F A096748 a(n) = Sum_{k=0..n} binomial(floor((n-k)/2), floor(k/2)). - _Paul Barry_, Jul 24 2004
%F A096748 a(n) = A079977(n) + A079977(n-2) + 2*A079977(n-1). - _R. J. Mathar_, Jul 15 2013
%t A096748 CoefficientList[Series[(1+x)^2/(1-x^2-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,0,1},{1,2,2,2},50] (* _Harvey P. Dale_, Jan 29 2012 *)
%Y A096748 Cf. A000045, A079977.
%Y A096748 Cf. A134972 and A239798 (limiting ratios for a(n+1)/a(n)).
%K A096748 easy,nonn
%O A096748 0,2
%A A096748 _Paul Barry_, Jul 07 2004