A147960 a(n) = ((9 + sqrt(2))^n + (9 - sqrt(2))^n)/2.

1, 9, 83, 783, 7537, 73809, 733139, 7365591, 74662657, 762046137, 7818480563, 80531005311, 831898131121, 8612216940609, 89299952572403, 927034007995143, 9631915890692737, 100138799400852969, 1041577033850627219

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A147959. 9th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..980
• Index entries for linear recurrences with constant coefficients, signature (18, -79).

Crossrefs

Cf. A077957, A098158, A147959.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r2)^n+(9-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

Mathematica

LinearRecurrence[{18, -79}, {1, 9}, 50] (* G. C. Greubel, Aug 17 2018 *)

PARI

x='x+O('x^30); Vec((1-9*x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

From Philippe Deléham, Nov 19 2008: (Start)

a(n) = 18*a(n-1) - 79*a(n-2), n > 1; a(0)=1, a(1)=9.

G.f.: (1 - 9*x)/(1 - 18*x + 79*x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*9^(2k)*2^(n-k))/9^n. (End)

E.g.f.: exp(9*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extensions

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