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A163350
a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
4
1, 6, 34, 188, 1028, 5592, 30344, 164464, 890896, 4824672, 26124832, 141453248, 765878336, 4146681216, 22451153024, 121555687168, 658129355008, 3563255219712, 19292230787584, 104452273224704, 565526954771456
OFFSET
0,2
COMMENTS
Binomial transform of A102285. Fourth binomial transform of A163403. Inverse binomial transform of A163346.
FORMULA
a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
a(n) = ((1+sqrt(2))*(4+sqrt(2))^n+(1-sqrt(2))*(4-sqrt(2))^n)/2.
G.f.: (1-2*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016
MATHEMATICA
LinearRecurrence[{8, -14}, {1, 6}, 30] (* Harvey P. Dale, May 08 2014 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009
(PARI) Vec((1-2*x)/(1-8*x+14*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009
EXTENSIONS
Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009
New name from G. C. Greubel, Dec 19 2016
STATUS
approved