A161938 a(n) = ((3+sqrt(2))*(2+sqrt(2))^n + (3-sqrt(2))*(2-sqrt(2))^n)/2.

3, 8, 26, 88, 300, 1024, 3496, 11936, 40752, 139136, 475040, 1621888, 5537472, 18906112, 64549504, 220385792, 752444160, 2569005056, 8771131904, 29946517504, 102243806208, 349082189824, 1191841146880, 4069200207872, 13893118537728, 47434073735168

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009, Jun 29 2009

Keywords

• nonn
• easy

Comments

Second binomial transform of A162255.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2).

Crossrefs

Cf. A162255, A135532, A083878.

Programs

GAP

a:=[3,8];; for n in [3..25] do a[n]:=4*a[n-1]-2*a[n-2]; od; a; # Muniru A Asiru, Sep 28 2018

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

Magma

I:=[3,8]; [n le 2 select I[n] else 4*Self(n-1) - 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 28 2018

Maple

seq(coeff(series((3-4*x)/(1-4*x+2*x^2),x,n+1), x, n), n = 0..25); # Muniru A Asiru, Sep 28 2018

Mathematica

CoefficientList[Series[(3-4*x)/(1-4*x+2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 28 2018 *)

PARI

my(x='x+O('x^50)); Vec((3-4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 28 2018

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 3; a(1) = 8.

G.f.: (3-4*x)/(1-4*x+2*x^2).

From G. C. Greubel, Sep 28 2018: (Start)

a(2*n) = 2^(n-1) * (Q(2*n +1) + 2*Q(2*n)), Q(m) = Pell-Lucas numbers = A002203(m).

a(2*n+1) = 2^(n-1) * (P(2*n+2) + 2*P(2*n+1)), P(m) = Pell numbers = A000129(m). (End)

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 01 2009