A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

1, 6, 38, 252, 1732, 12216, 87704, 637104, 4663312, 34298208, 253025888, 1870171584, 13839178816, 102484311936, 759279663488, 5626889356032, 41707163713792, 309171726460416, 2292017151256064, 16992367115418624, 125979822242317312, 934017384983574528, 6924894663564105728

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

6th binomial transform of A077957. Binomial transform of A083880. Inverse binomial transform of A147958. - Philippe Deléham, Nov 30 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-34).

Crossrefs

Cf. A077957, A083880, A098158, A147958.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+r2)^n+(6-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

Mathematica

LinearRecurrence[{12, -34}, {1, 6}, 50] (* G. C. Greubel, Aug 17 2018 *)

PARI

my(x='x+O('x^50)); Vec((1-6*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

From Philippe Deléham, Nov 19 2008: (Start)

a(n) = 12*a(n-1) - 34*a(n-2), n > 1; a(0)=1, a(1)=6.

G.f.: (1 - 6*x)/(1 - 12*x + 34*x^2).

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

E.g.f.: exp(6*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extensions

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