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 A166989 #20 Jan 10 2025 02:01:02
%S A166989 1,2,11,38,156,598,2353,9166,35843,139956,546792,2135796,8343205,
%T A166989 32590610,127308455,497301794,1942600788,7588340434,29642181517,
%U A166989 115790645854,452310642407,1766851828392,6901817263824,26960427965352
%N A166989 G.f.: A(x) = 1/(1 - 2*x - 7*x^2 - 2*x^3 + x^4).
%H A166989 G. C. Greubel, <a href="/A166989/b166989.txt">Table of n, a(n) for n = 0..1000</a>
%H A166989 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,2,-1).
%F A166989 G.f.: A(x) = exp( Sum_{n>=1} A000204(n)*A002203(n)*x^n/n ) where A000204 (Lucas numbers) forms the logarithmic derivative of the Fibonacci numbers (A000045) and A002203 forms the logarithmic derivative of the Pell numbers (A000129).
%F A166989 Recurrence: a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4) where a(k)=0 for k<0 with a(0)=1.
%F A166989 Radius of convergence: r = f*p where f=(sqrt(5)-1)/2, p=sqrt(2)-1:
%F A166989 (f*p-x)*(1/(f*p)-x)*(f/p+x)*(p/f+x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4.
%F A166989 For n >= 2, a(n) - a(n-2) = Fibonacci(n+1)*Pell(n+1) = A001582(n). - _Peter Bala_, Aug 30 2015
%F A166989 a(n) = (1/2) * Sum_{i=0..n+2} (-1)^(n-i+1) * Lucas(2*i-n-2) * Pell(i) * Pell(n+2-i). - _Vladimir Kruchinin_, Jan 10 2025
%t A166989 LinearRecurrence[{2, 7, 2, -1}, {1, 2, 11, 38}, 100] (* _G. C. Greubel_, May 30 2016 *)
%o A166989 (PARI) {a(n)=polcoeff(1/(1-2*x-7*x^2-2*x^3+x^4+x*O(x^n)),n)}
%o A166989 (PARI) {a(n)=if(n<0,0,if(n==0,1,2*a(n-1)+7*a(n-2)+2*a(n-3)-a(n-4)))}
%Y A166989 Cf. A000204, A000045, A002203, A000129, A001582.
%K A166989 nonn
%O A166989 0,2
%A A166989 _Paul D. Hanna_, Oct 26 2009