%I #20 Jan 25 2025 18:59:43
%S 1,12,113,984,8305,69156,572417,4725168,38957089,321004860,2644388561,
%T 21781512072,179402099473,1477598319444,12169714749665,
%U 100231029093216,825511191878977,6798972400658028,55996821859648049,461193717895377720
%N a(n) = ((6 + sqrt(5))^n - (6 - sqrt(5))^n)/(2*sqrt(5)).
%C Fourth binomial transform of A048877.
%C First differences are in A163146.
%C lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(5) = 8.236067977499789696....
%D S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H G. C. Greubel, <a href="/A153882/b153882.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12, -31).
%F From _Philippe Deléham_, Jan 03 2009: (Start)
%F a(n) = 12*a(n-1) - 31*a(n-2) for n>1, with a(0)=0, a(1)=1.
%F G.f.: x/(1 - 12*x + 31*x^2). (End)
%F a(n) = 12*a(n-1) - 31*a(n-2). - _G. C. Greubel_, Aug 31 2016
%t LinearRecurrence[{12, -31}, {1, 12}, 25] (* or *) Table[((6 + sqrt(5))^n - (6 - sqrt(5))^n)/(2*sqrt(5)) , {n,0,25}] (* _G. C. Greubel_, Aug 31 2016 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 04 2009
%o (Sage) [lucas_number1(n,12,31) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 27 2009
%o (Magma) I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-31*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 01 2016
%Y Cf. A002163 (decimal expansion of sqrt(5)), A048877, A163146.
%K nonn
%O 1,2
%A Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009
%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 04 2009
%E Edited by _Klaus Brockhaus_, Oct 11 2009