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 A239285 #60 Sep 08 2022 08:46:07
%S A239285 0,1,13,199,2977,44671,670033,10050559,150758257,2261374111,
%T A239285 33920611153,508809168319,7632137522737,114482062845151,
%U A239285 1717230942669073,25758464140052479,386376962100754417,5795654431511381791,86934816472670595793,1304022247090059199039
%N A239285 a(n) = (15^n - (-2)^n)/17.
%C A239285 Let k and t be positive integers and consider a(n) = k*a(n-1)+t*a(n-2) for n>=2, with a(0)=0, a(1)=1.
%C A239285 The roots of its characteristic equation are r1 = (k+sqrt(k^2+4t))/2 and r2 = (k-sqrt(k^2+4t))/2. Hence, the solution to the recurrence relation is the sequence {a(n)} where a(n) = alpha1*r1^n + alpha2*r2^n. It can be shown that alpha1 = 1/sqrt(k^2+4t) and alpha2 = -alpha1. It can be shown also that |r2/r1| < 1. Thus, a(n+1)/a(n) converges to r1 as n approaches infinity.
%C A239285 Note that limit a(n+1)/a(n) = 15 with k=13 and t=30.
%C A239285 If n > 20, then |a(n+1)/a(n) - 15| < 10^(-16).
%C A239285 Let b(n) be the number of strings of length n containing the 13-ary digits: 0,...,9,A,B,C or the 30 two-consecutive digits D0,D1,...,D9,DA,...,DT where A corresponds to 10, ..., T corresponds to 29. Then b(0)=1=a(2) and b(1)=13=a(3). The strings q_1q_2...q_n of length n can be partitioned into 2 groups A and B where A contains the strings where q_1=0,1,...,9,A,B,C and B contains the strings where q_1=D. Thus, |A|=13*b(n-1) and |B|=30*b(n-2). Hence, b(n) = 13*b(n-1) + 30*b(n-2) for n>1. Since b(0)=a(2) and b(1)=a(3), we can show that b(n) = a(n+2).
%H A239285 G. C. Greubel, <a href="/A239285/b239285.txt">Table of n, a(n) for n = 0..849</a>
%H A239285 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,30).
%F A239285 G.f.: x/(1 - 13*x - 30*x^2).
%F A239285 a(n) = 13*a(n-1) + 30*a(n-2) for n >= 2, a(0)=0, a(1)=1.
%F A239285 a(n) = (1/17)*(A001024(n) - A122803(n)), n >= 0.
%F A239285 a(0)=0, a(n) = Sum_{k=0..n-1} A001024(k)*A122803(n-1-k) for n > 0.
%F A239285 E.g.f.: (exp(15*x) - exp(-2*x))/17. - _G. C. Greubel_, May 26 2018
%t A239285 CoefficientList[Series[x/(1-13*x-30*x^2), {x,0,50}], x] (* or *) Table[
%t A239285 (15^n - (-2)^n)/17, {n,0,30}] (* or *) LinearRecurrence[{13,30}, {0,1}, 30] (* _G. C. Greubel_, May 26 2018 *)
%o A239285 (PARI) a(n) = (15^n-(-2)^n)/17; \\ _Michel Marcus_, Mar 16 2014
%o A239285 (PARI) x='x+O('x^30); concat([0], Vec()) \\ _G. C. Greubel_, May 26 2018
%o A239285 (Magma) [(15^n -(-2)^n)/17: n in [0..30]]; // _G. C. Greubel_, May 26 2018
%Y A239285 Cf. A001024, A122803.
%K A239285 nonn,easy
%O A239285 0,3
%A A239285 _Felix P. Muga II_, Mar 14 2014