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 A108851 #45 Aug 09 2025 16:58:46
%S A108851 1,2,11,50,233,1082,5027,23354,108497,504050,2341691,10878914,
%T A108851 50540729,234799658,1090820819,5067682250,23543191457,109375812578,
%U A108851 508132824683,2360658736466,10967033419913,50950109889050
%N A108851 a(n) = 4*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 2.
%C A108851 Binomial transform of A083098, second binomial transform of (1, 0, 7, 0, 49, 0, 243, 0, ...).
%H A108851 Vincenzo Librandi, <a href="/A108851/b108851.txt">Table of n, a(n) for n = 0..1000</a>
%H A108851 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,3).
%F A108851 a(n) = ((2 + sqrt(7))^n + (2 - sqrt(7))^n) / 2.
%F A108851 G.f.: (1 - 2*x) / (1 - 4*x - 3*x^2).
%F A108851 E.g.f.: exp(2*x)*cosh(sqrt(7)*x).
%F A108851 a(n+1)/a(n) converges to 2 + sqrt(7) = 4.645751311064...
%F A108851 Limit_{k->oo} a(n+k)/a(k) = A108851(n) + A015530(n)*sqrt(7); also lim_{n->oo} A108851(n)/A015530(n) = sqrt(7). - _Johannes W. Meijer_, Aug 01 2010
%F A108851 a(n) = Sum_{k=0..n} A201730(n,k)*6^k. - _Philippe Deléham_, Dec 06 2011
%F A108851 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013
%F A108851 a(n) = (2 + sqrt(7))^n - A015530(n)*sqrt(7). - _Robert FERREOL_, Aug 04 2025
%t A108851 LinearRecurrence[{4,3},{1,2},30] (* _Harvey P. Dale_, Jan 02 2022 *)
%o A108851 (Sage) [lucas_number2(n,4,-3)/2 for n in range(0, 22)] # _Zerinvary Lajos_, May 14 2009
%o A108851 (Magma) [Floor(((2 + Sqrt(7))^n + (2 - Sqrt(7))^n) / 2): n in [0..30]]; // _Vincenzo Librandi_, Jul 18 2011
%o A108851 (PARI) a(n)=round(((2+sqrt(7))^n+(2-sqrt(7))^n)/2) \\ _Charles R Greathouse IV_, Dec 06 2011
%Y A108851 Cf. A080042. - _Zerinvary Lajos_, May 14 2009
%Y A108851 Cf. A015530, A083098.
%Y A108851 Appears in A179596, A179597 and A126473. - _Johannes W. Meijer_, Aug 01 2010
%K A108851 easy,nonn
%O A108851 0,2
%A A108851 _Philippe Deléham_, Jul 11 2005