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

Author

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

Keywords

• nonn

Examples

			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
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1090
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (12,-31).

Programs

Magma

Z:= PolynomialRing(Integers()); N:=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

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 *)
LinearRecurrence[{12,-31},{1,6},30] (* Harvey P. Dale, Aug 28 2024 *)

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

Extensions

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

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