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A164303 a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 3, a(1) = 11. 3
3, 11, 25, 61, 147, 355, 857, 2069, 4995, 12059, 29113, 70285, 169683, 409651, 988985, 2387621, 5764227, 13916075, 33596377, 81108829, 195814035, 472736899, 1141287833, 2755312565, 6651912963, 16059138491, 38770189945, 93599518381 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Binomial transform of A164654. Inverse binomial transform of A164304.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..218 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (2,1).

FORMULA

a(n) = 2*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.

a(n) = ((3+4*sqrt(2))*(1+sqrt(2))^n + (3-4*sqrt(2))*(1-sqrt(2))^n)/2.

G.f.: (3+5*x)/(1-2*x-x^2).

E.g.f.: (3*cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(x). - G. C. Greubel, Sep 13 2017

MATHEMATICA

LinearRecurrence[{2, 1}, {3, 11}, 50] (* or *) CoefficientList[Series[(3 + 5*x)/(1 - 2*x - x^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 13 2017 *)

PROG

(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+4*r)*(1+r)^n+(3-4*r)*(1-r)^n)/2: n in [0..28] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009

(PARI) x='x+O('x^50); Vec((3+5*x)/(1-2*x-x^2)) \\ G. C. Greubel, Sep 13 2017

CROSSREFS

Cf. A164654, A164304.

Sequence in context: A212971 A258440 A184634 * A129082 A190476 A060746

Adjacent sequences:  A164300 A164301 A164302 * A164304 A164305 A164306

KEYWORD

nonn,easy,changed

AUTHOR

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

EXTENSIONS

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

STATUS

approved

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Last modified September 22 08:14 EDT 2017. Contains 292333 sequences.