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A163613
a(n) = ((1 + 3*sqrt(2))*(2 + sqrt(2))^n + (1 - 3*sqrt(2))*(2 - sqrt(2))^n)/2.
3
1, 8, 30, 104, 356, 1216, 4152, 14176, 48400, 165248, 564192, 1926272, 6576704, 22454272, 76663680, 261746176, 893657344, 3051137024, 10417233408, 35566659584, 121432171520, 414595366912, 1415517124608, 4832877764608
OFFSET
0,2
COMMENTS
Binomial transform of A048694. Second binomial transform of A163864. Inverse binomial transform of A163614.
FORMULA
a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
G.f.: (1+4*x)/(1-4*x+2*x^2).
E.g.f.: exp(2*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017
MATHEMATICA
LinearRecurrence[{4, -2}, {1, 8}, 50] (* G. C. Greubel, Jul 30 2017 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+r)^n+(1-3*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009
(PARI) x='x+O('x^50); Vec((1+4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 30 2017
CROSSREFS
Cf. A048694, A163864 (1, 6, 2, 12, 4, 24, ...), A163614.
Sequence in context: A107233 A098213 A372252 * A279217 A050477 A239612
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