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 A152107 #32 May 07 2025 09:41:09
%S A152107 1,6,41,306,2401,19326,157481,1290666,10606081,87262326,718359401,
%T A152107 5915180706,48713027041,401185722606,3304124833001,27212740595226,
%U A152107 224125017319681,1845905249384166,15202987455699881,125212786737489426
%N A152107 a(n) = ((6+sqrt(5))^n+(6-sqrt(5))^n)/2.
%H A152107 Robert Israel, <a href="/A152107/b152107.txt">Table of n, a(n) for n = 0..1090</a>
%H A152107 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
%H A152107 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-31).
%F A152107 From _Philippe Deléham_, Nov 26 2008: (Start)
%F A152107 a(n) = 12*a(n-1)-31*a(n-2), n>1 ; a(0)=1, a(1)=6 .
%F A152107 G.f.: (1-6*x)/(1-12*x+31*x^2).
%F A152107 a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*5^(n-k))/6^n. (End)
%F A152107 a(n) = Sum_{k=0..n} A027907(n,2k)*5^k . - _J. Conrad_, Aug 24 2016
%F A152107 E.g.f.: cosh(sqrt(5)*x)*exp(6*x). - _Ilya Gutkovskiy_, Aug 24 2016
%e A152107 For n=3, (6+sqrt(5))^3 = 216 + 108*sqrt(5) + 18*5 + 5*sqrt(5) = 306 + 113*sqrt(5) and (6-sqrt(5))^3 = 306 - 113*sqrt(5), so a(3) = (306 + 113*sqrt(5) + 306 - 113*sqrt(5))/2 = 306. - _Michael B. Porter_, Aug 25 2016
%p A152107 f:= gfun:-rectoproc({a(n)=12*a(n-1)-31*a(n-2),a(0)=1,a(1)=6},a(n),remember):
%p A152107 map(f, [$0..50]); # _Robert Israel_, Aug 25 2016
%t A152107 CoefficientList[Series[(1 - 6 x)/(1 - 12 x + 31 x^2), {x, 0, 19}], x] (* _Michael De Vlieger_, Aug 25 2016 *)
%t A152107 LinearRecurrence[{12,-31},{1,6},30] (* _Harvey P. Dale_, Aug 28 2024 *)
%o A152107 (Magma) Z<x>:= PolynomialRing(Integers()); N<r5>:=NumberField(x^2-5); S:=[ ((6+r5)^n+(6-r5)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 26 2008
%K A152107 nonn
%O A152107 0,2
%A A152107 Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008
%E A152107 Extended beyond a(6) by _Klaus Brockhaus_, Nov 26 2008
%E A152107 Typo in name corrected by _J. Conrad_, Aug 24 2016