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A147960
a(n) = ((9 + sqrt(2))^n + (9 - sqrt(2))^n)/2.
4
1, 9, 83, 783, 7537, 73809, 733139, 7365591, 74662657, 762046137, 7818480563, 80531005311, 831898131121, 8612216940609, 89299952572403, 927034007995143, 9631915890692737, 100138799400852969, 1041577033850627219
OFFSET
0,2
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
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
MATHEMATICA
LinearRecurrence[{18, -79}, {1, 9}, 50] (* G. C. Greubel, Aug 17 2018 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r2>:=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
(PARI) x='x+O('x^30); Vec((1-9*x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 17 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008
EXTENSIONS
Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008
STATUS
approved