A152107 a(n) = ((6+sqrt(5))^n+(6-sqrt(5))^n)/2.

1, 6, 41, 306, 2401, 19326, 157481, 1290666, 10606081, 87262326, 718359401, 5915180706, 48713027041, 401185722606, 3304124833001, 27212740595226, 224125017319681, 1845905249384166, 15202987455699881, 125212786737489426

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1090

Index entries for linear recurrences with constant coefficients, signature (12, -31).

Formula

From Philippe Deléham, Nov 26 2008: (Start)

a(n) = 12*a(n-1)-31*a(n-2), n>1 ; a(0)=1, a(1)=6 .

G.f.: (1-6*x)/(1-12*x+31*x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*5^(n-k))/6^n. (End)

a(n) = Sum_{k=0..n} A027907(n,2k)*5^k . - J. Conrad, Aug 24 2016

E.g.f.: cosh(sqrt(5)*x)*exp(6*x). - Ilya Gutkovskiy, Aug 24 2016

Example

For n=3, (6+sqrt(5))^3 = 216 + 108*sqrt(5) + 18*5 + 5*sqrt(5) = 306 + 113*sqrt(5) and (6-sqrt(5))^3 = 306 - 113*sqrt(5), so a(3) = (306 + 113*sqrt(5) + 306 - 113*sqrt(5))/2 = 306. - Michael B. Porter, Aug 25 2016

Maple

f:= gfun:-rectoproc({a(n)=12*a(n-1)-31*a(n-2), a(0)=1, a(1)=6}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Aug 25 2016

Mathematica

CoefficientList[Series[(1 - 6 x)/(1 - 12 x + 31 x^2), {x, 0, 19}], x] (* Michael De Vlieger, Aug 25 2016 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r5>:=NumberField(x^2-5); S:=[ ((6+r5)^n+(6-r5)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

Crossrefs

Sequence in context: A083067 A000402 A186654 * A143023 A078009 A127848

Adjacent sequences: A152104 A152105 A152106 * A152108 A152109 A152110

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

Typo in name corrected by J. Conrad, Aug 24 2016

Status

approved