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A015530 Expansion of x/(1-4*x-3*x^2). 23
0, 1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076, 4111843, 19102600, 88745929, 412291516, 1915403851, 8898489952, 41340171361, 192056155300, 892245135283, 4145149007032, 19257331433977, 89464772757004 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let b(1)=1, b(k)=floor(b(k-1))+3/b(k-1); then for n>1, b(n)=a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002

In general, x/(1-a*x-b*x^2) has a(n)=sum{k=0..floor((n-1)/2),C(n-k-1,k)b^k*a^(n-2k-1)}. - Paul Barry, Apr 23 2005

Pisano period lengths: 1, 2, 1, 4, 24, 2, 21, 4, 3, 24, 40, 4, 84, 42, 24, 8,288, 6, 18, 24,... - R. J. Mathar, Aug 10 2012

This is the Lucas sequence U(4,-3). - Bruno Berselli, Jan 09 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Wikipedia, Lucas sequence.

Index to sequences with linear recurrences with constant coefficients, signature (4,3).

FORMULA

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

a(n) = (A086901(n+2) - A086901(n+1))/6. - Ralf Stephan, Feb 01 2004

a(n) = sum{k=0..floor((n-1)/2), C(n-k-1, k)3^k*4^(n-2k-1)} - Paul Barry, Apr 23 2005

a(n) = ((2+sqrt(7))^n-(2-sqrt(7))^n)/sqrt(28). Offset 1. a(3)=19 [From Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009]

Contribution from Johannes W. Meijer, Aug 01 2010: (Start)

Limit(a(n+k)/a(k), k=infinity) = A108851(n)+a(n)*sqrt(7)

Limit(A108851(n)/a(n), n=infinity) = sqrt(7)

(End)

G.f. x*G(0) where G(k)= 1 + (4*x+3*x^2)/(1 - (4*x+3*x^2)/(4*x + 3*x^2 + 1/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 28 2012

G.f.: G(0)*x/(2-4*x), where G(k)= 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013

MATHEMATICA

a[n_]:=(MatrixPower[{{1, 6}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, -1, 40}] [From Vladimir Joseph Stephan Orlovsky, Feb 20 2010]

LinearRecurrence[{4, 3}, {0, 1}, 30] (* Vincenzo Librandi, Jun 19 2012 *)

PROG

(Sage) [lucas_number1(n, 4, -3) for n in xrange(0, 23)]# [From Zerinvary Lajos, Apr 23 2009]

(MAGMA) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012

CROSSREFS

Appears in A179596, A126473 and A179597. [Johannes W. Meijer, Aug 01 2010]

Cf. A080042: Lucas sequence V(4,-3).

Sequence in context: A017962 A192526 A084155 * A181880 A010907 A229242

Adjacent sequences:  A015527 A015528 A015529 * A015531 A015532 A015533

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified November 28 20:22 EST 2014. Contains 250399 sequences.