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 A065874 #35 Dec 27 2024 10:48:00
%S A065874 1,1,43,85,1891,5461,84883,314245,3879331,17077621,180009523,
%T A065874 897269605,8457669571,46142992981,401365114963,2339370820165,
%U A065874 19196705648611,117450280095541,923711917337203,5856623681349925,44652524209512451,290630718826209301,2166036735625732243
%N A065874 a(n) = (7^(n+1) - (-6)^(n+1))/13.
%C A065874 A second-order recurrence of promic type (integer roots).
%C A065874 If the number j = A002378(m) is promic (= i(i+1)), then a(n) = a(n-1) + j*a(n-2), a(0) = a(1) = 1 has a closed-form solution involving only powers of integers. The binomial coefficient sum solves the recurrence regardless of promicity (cf. GKP reference).
%C A065874 Hankel transform is := 1,42,0,0,0,0,0,0,0,0,0,0,... - _Philippe Deléham_, Nov 02 2008
%D A065874 R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, p. 204.
%H A065874 Harry J. Smith, <a href="/A065874/b065874.txt">Table of n, a(n) for n = 0..150</a>
%H A065874 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,42).
%F A065874 a(n) = a(n-1) + 42a(n-2); a(0) = a(1) = 1.
%F A065874 G.f.: -1/((6*x+1)*(7*x-1)). - _R. J. Mathar_, Nov 16 2007
%p A065874 n->sum(binomial(n-k, k)*(42)^k, k=0..n)
%t A065874 LinearRecurrence[{1,42},{1,1},30] (* _Harvey P. Dale_, Apr 30 2017 *)
%o A065874 (PARI) a(n) = { (7^(n+1) - (-6)^(n+1))/13 } \\ _Harry J. Smith_, Nov 02 2009
%Y A065874 Cf. A001045 (j=2), A015441 (j=6), A053404 (j=12), A053428 (j=20), A053430 (j=30).
%K A065874 nonn,easy
%O A065874 0,3
%A A065874 _Len Smiley_, Dec 07 2001