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A054486 Expansion of (1+2x)/(1-3x+x^2). 8
1, 5, 14, 37, 97, 254, 665, 1741, 4558, 11933, 31241, 81790, 214129, 560597, 1467662, 3842389, 10059505, 26336126, 68948873, 180510493, 472582606, 1237237325, 3239129369, 8480150782, 22201322977, 58123818149, 152170131470, 398386576261, 1042989597313 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A000285. - R. J. Mathar, Oct 26 2011

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

LINKS

Table of n, a(n) for n=0..28.

I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (3,-1).

FORMULA

a(n) = 3a(n-1) - a(n-2), a(0)=1, a(1)=5.

a(n) + 7*A001519(n) = A005248(n). - Creighton Dement, Oct 30 2004

a(n) = Lucas(2n+1) + Fibonacci(2n) = A002878(n) + A001906(n) = A025169(n-1) + A001906(n+1).

a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-6)^k. - Philippe Deléham, Mar 05 2014

0 = -11 + a(n)^2 - 3*a(n)*a(n+1) + a(n+1)^2 for all n in Z. - Michael Somos, Mar 17 2015

a(n) = -2*F(n)^2 + 6*F(n)*F(n+1) + F(n+1)^2 for all n in Z where F = Fibonacci. - Michael Somos, Mar 17 2015

a(n) = 3*F(2*n) + F(2*n+1) for all n in Z where F = Fibonacci. - Michael Somos, Mar 17 2015

a(n) = -A100545(-2-n) for all n in Z. - Michael Somos, Mar 17 2015

a(n) = A000285(2*n) = A228208(2*n+1) = A104449(2*n+1) for all n in Z. - Michael Somos, Mar 17 2015

EXAMPLE

a(n) = (5*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - (((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)))/sqrt(5).

G.f. = 1 + 5*x + 14*x^2 + 37*x^3 + 97*x^4 + 254*x^5 + 665*x^6 + 1741*x^7 + ...

MATHEMATICA

CoefficientList[Series[(2*z + 1)/(z^2 - 3*z + 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

a[ n_] := 3 Fibonacci[2 n] + Fibonacci[2 n + 1]; (* Michael Somos, Mar 17 2015 *)

PROG

(PARI) Vec((1+2*x)/(1-3*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 15 2011

(PARI) {a(n) = 3*fibonacci(2*n) + fibonacci(2*n+1)}; /* Michael Somos, Mar 17 2015 */

CROSSREFS

Cf. A000285, A002878, A100545, A104449, A228208.

Sequence in context: A292170 A224716 A127980 * A072130 A196976 A045553

Adjacent sequences:  A054483 A054484 A054485 * A054487 A054488 A054489

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, May 06 2000

EXTENSIONS

"a(1)=5", not "a(0)=5" Dan Nielsen (nielsed(AT)uah.edu), Sep 10 2009

STATUS

approved

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Last modified November 20 00:42 EST 2017. Contains 294957 sequences.