A164031 a(n) = ((2+3*sqrt(2))*(5+sqrt(2))^n+(2-3*sqrt(2))*(5-sqrt(2))^n)/4.

1, 8, 57, 386, 2549, 16612, 107493, 692854, 4456201, 28626368, 183771057, 1179304106, 7566306749, 48539073052, 311365675293, 1997258072734, 12811170195601, 82174766283128, 527090748332457, 3380887858812626

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Keywords

• nonn

Comments

Binomial transform of A164072. Fifth binomial transform of A164073.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-23).

Crossrefs

Cf. A164072, A164073 (1, 3, 2, 6, 4, 12).

Programs

GAP

a:=[1,8];; for n in [3..25] do a[n]:=10*a[n-1]-23*a[n-2]; od; a; # Muniru A Asiru, Apr 04 2018

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+3*r)*(5+r)^n+(2-3*r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 09 2009

Mathematica

CoefficientList[Series[(1 - 2*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{10,-23}, {1,8}, 50] (* G. C. Greubel, Sep 07 2017 *)

PARI

x='x+O('x^50); Vec((1-2*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 07 2017

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

G.f.: (1-2*x)/(1-10*x+23*x^2).

E.g.f.: exp(5*x)*(2*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/2. - G. C. Greubel, Apr 03 2018

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 09 2009