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 A089508 #19 Sep 08 2022 08:45:12
%S A089508 1,14,103,713,4894,33551,229969,1576238,10803703,74049689,507544126,
%T A089508 3478759199,23843770273,163427632718,1120149658759,7677619978601,
%U A089508 52623190191454,360684711361583,2472169789339633,16944503814015854
%N A089508 Solution to a binomial problem together with companion sequence A081016(n-1).
%C A089508 a(n) and b(n) := A081016(n-1) are the solutions to the Diophantine equation binomial(a,b) = binomial(a+1,b-1). The first few binomials are given by A090162(n).
%D A089508 A. I. Shirshov: On the equation binomial(n,m)=binomial(n+1,m-1), pp. 83-86, in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999.
%H A089508 G. C. Greubel, <a href="/A089508/b089508.txt">Table of n, a(n) for n = 1..1000</a>
%H A089508 Wikipedia, <a href="http://en.wikipedia.org/wiki/Singmaster's_conjecture">Singmaster's conjecture</a>
%H A089508 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8, -8, 1).
%F A089508 G.f.: x*(1+6*x-x^2)/((1-x)*(1-7*x+x^2)).
%F A089508 a(n) = A081018(n) - 1 = F(2*n)*F(2*n+1) - 1, n>=1; with F(n) := A000045(n) (Fibonacci).
%e A089508 n = 2: a(2) = 14, b(2) = A081016(1) = 6 satisfy binomial(14,6) = 3003 = binomial(15,5). 3003 = A090162(2).
%t A089508 Rest[CoefficientList[Series[x*(1 + 6*x - x^2)/((1 - x)*(1 - 7*x + x^2)), {x, 0, 50}], x]] (* _G. C. Greubel_, Dec 18 2017 *)
%o A089508 (PARI) x='x+O('x^30); Vec(x*(1 + 6*x - x^2)/((1 - x)*(1 - 7*x + x^2))) \\ _G. C. Greubel_, Dec 18 2017
%o A089508 (Magma) [Fibonacci(2*n)*Fibonacci(2*n+1) - 1: n in [1..30]]; // _G. C. Greubel_, Dec 18 2017
%Y A089508 Equals A081018 - 1.
%K A089508 nonn,easy
%O A089508 1,2
%A A089508 _Wolfdieter Lang_, Dec 01 2003