A162563 a(n) = ((5+sqrt(3))*(2+sqrt(3))^n + (5-sqrt(3))*(2-sqrt(3))^n)/2.

5, 13, 47, 175, 653, 2437, 9095, 33943, 126677, 472765, 1764383, 6584767, 24574685, 91713973, 342281207, 1277410855, 4767362213, 17792037997, 66400789775, 247811121103, 924843694637, 3451563657445, 12881410935143, 48074080083127

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

Keywords

• nonn

Comments

Binomial transform of A162562. Second binomial transform of A162813. Inverse binomial transform of A162814.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-1).

Crossrefs

Cf. A162562, A162813, A162814.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r)*(2+r)^n+(5-r)*(2-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 14 2009

Mathematica

LinearRecurrence[{4,-1},{5,13},30] (* Harvey P. Dale, Aug 25 2014 *)

Formula

a(n) = 4*a(n-1) - a(n-2) for n > 1; a(0) = 5, a(1) = 13.

G.f.: (5-7*x)/(1-4*x+x^2).

a(n) = 4*a(n-1) - a(n-2), with a(0)=5 and a(1)=13. - Paolo P. Lava, Jul 15 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 18 2009