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A085151 Numbers generated by the Fibonacci polynomial x^4 + 3x^2 + 1.

%I A085151 #21 Jun 05 2025 00:42:58
%S A085151 5,29,109,305,701,1405,2549,4289,6805,10301,15005,21169,29069,39005,
%T A085151 51301,66305,84389,105949,131405,161201,195805,235709,281429,333505,
%U A085151 392501,459005,533629,617009,709805,812701,926405,1051649,1189189
%N A085151 Numbers generated by the Fibonacci polynomial x^4 + 3x^2 + 1.
%C A085151 Start with the Fibonacci polynomials of A011973 (see "examples") and put in appropriate exponents, e.g. {1,1} = x^2 + 1, the generator of A002522; {1,2} = x^3 + 2x, the generator of A054602; and to get the next polynomial, multiply by x and add the previous polynomial, such that the generator for A085151 = x^4 + 3x^2 + 1 = (x)(x^3+2x) + (x^2+1).
%H A085151 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A085151 a(n) = n^4 + 3*n^2 + 1.
%F A085151 a(n) = n*A054602(n) + A002522(n).
%F A085151 a(n) = denominator of [n, n, n, n]; with numerator = A054602(n).
%F A085151 a(n) = A057721(n). - _R. J. Mathar_, Sep 12 2008
%F A085151 From _Chai Wah Wu_, May 20 2025: (Start)
%F A085151 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
%F A085151 G.f.: x*(-x^4 - 14*x^2 - 4*x - 5)/(x - 1)^5. (End)
%F A085151 Sum_{n>=1} 1/a(n) = - 1/2 - (Pi/10)*((5*sinh(Pi)+sqrt(5)*sinh(sqrt(5)*Pi))/(cosh(Pi)-cosh(sqrt(5)*Pi))). - _Amiram Eldar_, Jun 04 2025
%e A085151 a(2) = f(2) of x^4 + 3x^2 + 1 = 29.
%e A085151 a(2) = 29 = (2)*A054602(2) + A002522(2) = (2)(12) + 5.
%e A085151 [2,2,2,2] = 12/29; a(2) = 29, & 12 = A054602(2). Thus [n,n,n,n] = A054602(n)/A085151(n).
%t A085151 a[n_] := n^4 + 3n^2 + 1; Array[a, 33]
%o A085151 (Python)
%o A085151 def A085151(n): return (m:=n**2)*(m+3)+1 # _Chai Wah Wu_, May 20 2025
%Y A085151 Cf. A002522, A011973, A054602, A057721, A085151.
%K A085151 nonn
%O A085151 1,1
%A A085151 _Gary W. Adamson_, Jun 21 2003
%E A085151 More terms from _Robert G. Wilson v_, Aug 06 2006