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A247564 a(n) = 3*a(n-2) - 4*a(n-4) with a(0) = 2, a(1) = 1, a(2) = 3, a(3) = 1. 3
2, 1, 3, 1, 1, -1, -9, -7, -31, -17, -57, -23, -47, -1, 87, 89, 449, 271, 999, 457, 1201, 287, -393, -967, -5983, -4049, -16377, -8279, -25199, -8641, -10089, 7193, 70529, 56143, 251943, 139657, 473713, 194399, 413367, 24569, -654751, -703889, -3617721 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-4).

FORMULA

G.f.: (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4).

a(n) = A247487(n) * 3^( n == 1 (mod 4) ) for all n in Z.

a(2*n) = A247563(n). a(2*n + 1) = A247560(n).

0 = a(n)*(+2*a(n+2)) + a(n+1)*(+2*a(n+1) - 8*a(n+2) + a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z.

a(n) = C*(3/2-A)^D*(3/2-A)^B*(-1)^n*(3/2-A)^(-1/4)*sqrt(7) - C*(-1)^n*(3/2+A)^B*(3/2+A)^D*sqrt(7)*(3/2+A)^(-1/4) + B*(3/2+A)^B*(3/2+A)^D*(3/2+A)^(-1/4) + C*(3/2+A)^B*(3/2+A)^D*sqrt(7)*(3/2+A)^(-1/4) + 3/4*(3/2+A)^B*(3/2+A)^D*(3/2+A)^(-1/4) + (1/4)*(3/2-A)^D*(3/2-A)^B*(-1)^n*(3/2-A)^(-1/4) - C*(3/2-A)^D*(3/2-A)^B*(3/2-A)^(-1/4)*sqrt(7) + (3/4)*(3/2-A)^D*(3/2-A)^B*(3/2-A)^(-1/4), with A=(1/2)*i*sqrt(7), B=(1/4)*(-1)^n, C=(1/28)*i, D=(1/2)*n. - Paolo P. Lava, Sep 22 2014

EXAMPLE

G.f. = 2 + x + 3*x^2 + x^3 + x^4 - x^5 - 9*x^6 - 7*x^7 - 31*x^8 - 17*x^9 + ...

MATHEMATICA

CoefficientList[Series[(2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 04 2018 *)

PROG

(PARI) {a(n) = if( n<0, n=-n; 2^-n, 1) * polcoeff( (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4) + x * O(x^n), n)};

(Haskell)

a247564 n = a247564_list !! n

a247564_list = [2, 1, 3, 1] ++ zipWith (-) (map (* 3) $ drop 2 a247564_list)

                                        (map (* 4) $ a247564_list)

-- Reinhard Zumkeller, Sep 20 2014

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4)));  // G. C. Greubel, Aug 04 2018

CROSSREFS

Cf. A247487, A247560, A247563.

Sequence in context: A119804 A300977 A144869 * A193870 A058564 A226006

Adjacent sequences:  A247561 A247562 A247563 * A247565 A247566 A247567

KEYWORD

sign,easy

AUTHOR

Michael Somos, Sep 20 2014

STATUS

approved

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Last modified June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)