login
A163614
a(n) = ((1 + 3*sqrt(2))*(3 + sqrt(2))^n + (1 - 3*sqrt(2))*(3 - sqrt(2))^n)/2.
4
1, 9, 47, 219, 985, 4377, 19367, 85563, 377809, 1667913, 7362815, 32501499, 143469289, 633305241, 2795546423, 12340141851, 54472026145, 240451163913, 1061402800463, 4685258655387, 20681732329081, 91293583386777
OFFSET
0,2
COMMENTS
Binomial transform of A163613. Third binomial transform of A163864. Inverse binomial transform of A163615.
FORMULA
a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1+3*x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017
MATHEMATICA
LinearRecurrence[{6, -7}, {1, 9}, 30] (* Harvey P. Dale, Sep 24 2015 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+3*r)*(3+r)^n+(1-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
(PARI) x='x+O('x^50); Vec((1+3*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 30 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009
EXTENSIONS
Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009
STATUS
approved