A164300 a(n) = ((1+4*sqrt(2))*(4+sqrt(2))^n + (1-4*sqrt(2))*(4-sqrt(2))^n)/2.

1, 12, 82, 488, 2756, 15216, 83144, 452128, 2453008, 13294272, 72012064, 389976704, 2111644736, 11433484032, 61904845952, 335169991168, 1814692086016, 9825156811776, 53195565289984, 288012326955008

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A164299. Fourth binomial transform of A164587. Inverse binomial transform of A164301.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-14).

Crossrefs

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), this sequence (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081180, A083879, A164587.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+r)^n+(1-4*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

Mathematica

LinearRecurrence[{8,-14},{1,12},30] (* Harvey P. Dale, Apr 13 2012 *)

PARI

my(x='x+O('x^50)); Vec((1+4*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Sep 13 2017

Sage

[( (1+4*x)/(1-8*x+14*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

Formula

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 12.

G.f.: (1+4*x)/(1-8*x+14*x^2).

E.g.f.: (4*sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(4*x). - Ilya Gutkovskiy, Jun 24 2016

From G. C. Greubel, Mar 12 2021: (Start)

a(n) = 2*A083879(n) + 8*A081180(n).

a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*3^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009