A154346 a(n) = 12*a(n-1) - 28*a(n-2) for n > 1, with a(0)=1, a(1)=12.

1, 12, 116, 1056, 9424, 83520, 738368, 6521856, 57587968, 508443648, 4488860672, 39629905920, 349870772224, 3088811900928, 27269361188864, 240745601040384, 2125405099196416, 18763984361226240, 165656469557215232

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

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.

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

Links

• R. J. Mathar, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (12, -28).

Crossrefs

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

Programs

Magma

Z:=PolynomialRing(Integers()); N:=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

Mathematica

Join[{a=1,b=12},Table[c=12*b-28*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
LinearRecurrence[{12,-28},{1,12},20] (* Harvey P. Dale, May 23 2012 *)
Rest@ CoefficientList[Series[x/(1 - 12 x + 28 x^2), {x, 0, 19}], x] (* Michael De Vlieger, Sep 13 2016 *)

PARI

a(n)=([0,1; -28,12]^(n-1)*[1;12])[1,1] \\ Charles R Greathouse IV, Sep 13 2016

Sage

[lucas_number1(n,12,28) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = 12*a(n-1) - 28*a(n-2) for n > 1. - Philippe Deléham, Jan 12 2009

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

G.f.: x/(1 - 12*x + 28*x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 08 2009

E.g.f.: (1/(2*sqrt(2)))*exp(6*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016

a(n) =2^(n-1)*A081179(n). - R. J. Mathar, Feb 04 2021

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 12 2009

Edited by Klaus Brockhaus, Oct 08 2009

Offset corrected. - R. J. Mathar, Jun 19 2021