A147962 a(n) = ((7+sqrt(3))^n + (7-sqrt(3))^n) / 2.

1, 7, 52, 406, 3292, 27412, 232336, 1991752, 17197072, 149138416, 1296872512, 11295848032, 98485736896, 859191307072, 7498334401792, 65453881499776, 571430958514432, 4989154870212352, 43562344091309056, 380371693248558592, 3321335877279603712

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (14, -46).

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((7+r3)^n+(7-r3)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

Mathematica

LinearRecurrence[{14,-46},{1,7},30] (* Harvey P. Dale, Aug 18 2012 *)

PARI

Vec((1-7*x) / (1-14*x+46*x^2) + O(x^30)) \\ Colin Barker, Sep 24 2017

Formula

From Zak Seidov, Nov 19 2008: (Start)

a(0) = 1; a(1) = 7; a(n) = 14 * a(n - 1) - 46 * a(n - 2);

a(n) = ((7+sqrt(3))^n + (7-sqrt(3))^n)/2. (End)

From Philippe Deléham, Nov 19 2008: (Start)

G.f.: (1-7*x)/(1-14*x+46*x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2*k)*3^(n-k))/7^n. (End)

Extensions

Extended beyond a(6) by Klaus Brockhaus and Zak Seidov, Nov 19 2008