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Expansion of (1+x)^2/(1-x^2-x^4).
5

%I #43 Apr 20 2024 10:28:30

%S 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 377,466,610,754,987,1220,1597,1974,2584,3194,4181,5168,6765,8362,

%U 10946,13530,17711,21892,28657,35422,46368,57314,75025,92736,121393,150050

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

%C 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 Harvey P. Dale, <a href="/A096748/b096748.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).

%F a(n) = a(n-2) + a(n-4).

%F a(n) = 2*F((n+1)/2)*(1-(-1)^n)/2 + F((n+4)/2)*(1+(-1)^n)/2.

%F a(2*n) = A000045(n+2); a(2*n+1) = 2*A000045(n+1).

%F a(n) = Sum_{k=0..n} binomial(floor((n-k)/2), floor(k/2)). - _Paul Barry_, Jul 24 2004

%F a(n) = A079977(n) + A079977(n-2) + 2*A079977(n-1). - _R. J. Mathar_, Jul 15 2013

%t 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 Cf. A000045, A079977.

%Y Cf. A134972 and A239798 (limiting ratios for a(n+1)/a(n)).

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 07 2004