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 A344684 #23 Aug 28 2021 04:47:34
%S A344684 1,3,12,46,181,705,2757,10765,42058,164280,641739,2506789,9792253,
%T A344684 38251227,149420064,583676434,2280003517,8906330973,34790619369,
%U A344684 135901886149,530870766310
%N A344684 Sum of two consecutive products of Fibonacci and Pell numbers: F(n)*P(n) + F(n+1)*P(n+1).
%C A344684 a(n) is the numerator of the continued fraction [1,...,1,2,...,2] with n 1's followed by n 2's.
%H A344684 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,2,-1).
%F A344684 a(n) = F(n)*P(n) + F(n+1)*P(n+1) for F(n) = A000045(n) the Fibonacci numbers and P(n) = A000129(n) the Pell numbers.
%F A344684 a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
%F A344684 G.f.: (1 + x - x^2 - x^3)/(1 - 2*x - 7*x^2 - 2*x^3 + x^4).
%F A344684 a(n) = A001582(n-1) + A001582(n) for n >= 1.
%e A344684 For n=3, a(3)=46 which is F(3)*P(3) + F(4)*P(4) = 2*5 + 3*12 = 46. Also, the continued fraction [1,1,1,2,2,2] with 3 1's followed by 3 2's has numerator 46.
%t A344684 Table[Fibonacci[n] Fibonacci[n, 2] + Fibonacci[n + 1] Fibonacci[n + 1, 2], {n, 0, 30}]
%o A344684 (PARI) P(n) = ([2, 1; 1, 0]^n)[2, 1]; \\ A000129
%o A344684 a(n) = fibonacci(n)*P(n)+ fibonacci(n+1)*P(n+1); \\ _Michel Marcus_, Aug 18 2021
%Y A344684 Cf. A000045, A000129, A001582.
%K A344684 nonn,easy
%O A344684 0,2
%A A344684 _Greg Dresden_ and _Hexuan Wang_, Aug 17 2021