A154346 - OEIS

%I #38 Sep 08 2022 08:45:40

%S 1,12,116,1056,9424,83520,738368,6521856,57587968,508443648,

%T 4488860672,39629905920,349870772224,3088811900928,27269361188864,

%U 240745601040384,2125405099196416,18763984361226240,165656469557215232

%N a(n) = 12*a(n-1) - 28*a(n-2) for n > 1, with a(0)=1, a(1)=12.

%C Binomial transform of A164547, second binomial transform of A164546, third binomial transform of A038761, fourth binomial transform of A164545, fifth binomial transform of A164544, sixth binomial transform of A164640.

%C Lim_{n -> infinity} a(n)/a(n-1) = 6 + 2*sqrt(2) = 8.8284271247....

%H R. J. Mathar, <a href="/A154346/b154346.txt">Table of n, a(n) for n = 0..100</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12, -28).

%F a(n) = 12*a(n-1) - 28*a(n-2) for n > 1. - _Philippe Deléham_, Jan 12 2009

%F a(n) = ( (6 + 2*sqrt(2))^n - (6 - 2*sqrt(2))^n )/(4*sqrt(2)).

%F G.f.: x/(1 - 12*x + 28*x^2). - _Klaus Brockhaus_, Jan 12 2009, corrected Oct 08 2009

%F E.g.f.: (1/(2*sqrt(2)))*exp(6*x)*sinh(2*sqrt(2)*x). - _G. C. Greubel_, Sep 13 2016

%F a(n) =2^(n-1)*A081179(n). - _R. J. Mathar_, Feb 04 2021

%t Join[{a=1,b=12},Table[c=12*b-28*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2011 *)

%t LinearRecurrence[{12,-28},{1,12},20] (* _Harvey P. Dale_, May 23 2012 *)

%t Rest@ CoefficientList[Series[x/(1 - 12 x + 28 x^2), {x, 0, 19}], x] (* _Michael De Vlieger_, Sep 13 2016 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((6+2*r)^n-(6-2*r)^n)/(4*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 12 2009

%o (Sage) [lucas_number1(n,12,28) for n in range(1, 20)] # _Zerinvary Lajos_, Apr 27 2009

%o (PARI) a(n)=([0,1; -28,12]^(n-1)*[1;12])[1,1] \\ _Charles R Greathouse IV_, Sep 13 2016

%Y Cf. A002193 (decimal expansion of sqrt(2)), A164547, A164546, A038761, A164545, A164544, A164640.

%K nonn,easy

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 12 2009

%E Edited by _Klaus Brockhaus_, Oct 08 2009

%E Offset corrected. - _R. J. Mathar_, Jun 19 2021