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A138573 a(0)=0, a(1)=1, a(2)=2, a(3)=5, a(n+4) = 2a(n+3) + 2a(n+2) + 2a(n+1) - a(n). 4
0, 1, 2, 5, 16, 45, 130, 377, 1088, 3145, 9090, 26269, 75920, 219413, 634114, 1832625, 5296384, 15306833, 44237570, 127848949, 369490320, 1067846845, 3086134658, 8919094697, 25776662080, 74495936025, 215297250946, 622220603405 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 23 2008

Case P1 = 2, P2 = -4, Q = 1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 04 2014

LINKS

Table of n, a(n) for n=0..27.

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index to divisibility sequences

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

FORMULA

a(n) = round(w^n/2/sqrt(5)) where w=(1+r)/(1-r)=2.89005363826396... and r=sqrt(sqrt(5)-2)=0.485868271756...; For n>=3 a(n)=A071101(n+3)

G.f.: -x*(x-1)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4). - R. J. Mathar, Jun 03 2009

From Peter Bala, Mar 04 2014: (Start)

Define a Lucas sequence {U(n)} in the ring of Gaussian integers by the recurrence U(n) = (1 + i)*U(n-1) + U(n-2) with U(0) = 0 and U(1) = 1. Then a(n) = |U(n)|^2.

Let a, b denote the zeros of x^2 - (1 + i)*x - 1 and c, d denote the zeros of x^2 - (1 - i)*x - 1.

Then a(n) = (a^n - b^n)*(c^n - d^n)/((a - b)*(c - d)).

a(n) = (alpha(1)^n + beta(1)^n - alpha(2)^n - beta(2)^n)/(2*sqrt(5)), where alpha(1), beta(1) are the roots of x^2 - ( 1 + sqrt(5))*x + 1 = 0, and alpha(2), beta(2) are the roots of x^2 - (1 - sqrt(5))*x + 1 = 0.

The o.g.f. is the Hadamard product of the rational functions x/(1 - (1 + i)x - x^2) and x/(1 - (1 - i)x - x^2). (End)

From Peter Bala, Mar 24 2014: (Start)

a(n) = (1/sqrt(5))*(T(n,phi) - T(n,-1/phi)), where phi = 1/2*(1 + sqrt(5)) is the golden ratio and T(n,x) denotes the Chebyshev polynomial of the first kind. Compare with the Fibonacci numbers, A000045, whose terms are given by the Binet formula 1/sqrt(5)*( phi^n - (-1/phi)^n ).

a(n) = top right (or bottom left) entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1]; the off-diagonal elements of M^n give the sequence of Fibonacci numbers. Bottom right entry of the matrix T(n, M) gives A138574. See the remarks in A100047 for the general connection between Chebyshev polynomials and linear divisibility sequences of the fourth order. (End)

a(n) = (((phi + sqrt(phi))^n + (phi - sqrt(phi))^n)/2 - (-1)^n * cos(n*arctan(sqrt(phi))))/sqrt(5), where phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, May 11 2016

a(n) = A143056(n+1)^2 + A272665(n+1)^2. - Vladimir Reshetnikov, Oct 05 2016

MATHEMATICA

Round@Table[(((GoldenRatio + Sqrt[GoldenRatio])^n + (GoldenRatio - Sqrt[GoldenRatio])^n)/2 - (-1)^n Cos[n ArcTan[Sqrt[GoldenRatio]]])/Sqrt[5], {n, 0, 20}] (* or *) LinearRecurrence[{2, 2, 2, -1}, {0, 1, 2, 5}, 20] (* Vladimir Reshetnikov, May 11 2016 *)

Table[Abs[Fibonacci[n, 1 + I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 05 2016 *)

CROSSREFS

Cf. A071101, A000045, A100047, A138574, A143056, A272665.

Sequence in context: A148374 A182884 A152428 * A148375 A075887 A148376

Adjacent sequences:  A138570 A138571 A138572 * A138574 A138575 A138576

KEYWORD

nonn

AUTHOR

Benoit Cloitre, May 12 2008

STATUS

approved

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Last modified May 30 04:27 EDT 2017. Contains 287305 sequences.