A105309 a(n) = |b(n)|^2 = x^2 + 3*y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,c,c,c) where c = 1/sqrt(3).

1, 1, 2, 5, 9, 20, 41, 85, 178, 369, 769, 1600, 3329, 6929, 14418, 30005, 62441, 129940, 270409, 562725, 1171042, 2436961, 5071361, 10553600, 21962241, 45703841, 95110562, 197926885, 411889609, 857150100, 1783745641, 3712008565

Offset

0,3

Comments

Prepending 0 and keeping the offset at 0, turns this into a divisibility sequence with g.f. x(1-x^2)/(1-x-2x^2-x^3+x^4). - T. D. Noe, Dec 22 2008

Equals INVERT transform of (1, 1, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Apr 28 2009

Sequence gives the norm of the coefficients in 1/(1 - I*x - I*x^2), where I^2=-1. - Paul D. Hanna, Dec 06 2011

This is the case P1 = 1, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 27 2014

Links

Table of n, a(n) for n=0..31.

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

Ricky X. F. Chen and Louis W. Shapiro, On Sequences G(n) satisfying G(n) = (d+2)G(n-1)-G(n-2), J. Int. Seq. 10 (2007) 07.8.1, Theorem 16.

Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 8.

Eric Weisstein's World of Mathematics, Quaternion

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

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

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

Formula

a(n) = A092886(n+1) - A092886(n-1), n > 0.

a(n) = A201837(n)^2 + A201838(n)^2. - Paul D. Hanna, Dec 06 2011

From Peter Bala, Mar 27 2014: (Start)

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(17))/4 and beta = (1 - sqrt(17))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1/2].

a(n) = U(n-1,(1 + i)/sqrt(8))*U(n-1,(1 - i)/sqrt(8)), where U(n,x) denotes the Chebyshev polynomial of the second kind.

The o.g.f. is the Chebyshev transform of the rational function x/(1 - x + 4*x^2) = x + x^2 + 5*x^2 + 9*x^4 + 29*x^5 + ... (see A006131), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)).

See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

a(n) = abs(((sqrt(4*i - 1) + i)^(n+1) - (i - sqrt(4*i - 1))^(n+1)) / 2^(n+1) / sqrt(4*i - 1))^2. - Daniel Suteu, Dec 20 2016

a(n) = a(-2-n) for all n in Z. - Michael Somos, Dec 20 2016

G.f.: (1+x)*(1-x)/(1-x-2*x^2-x^3+x^4). - R. J. Mathar, Apr 26 2024

Example

G.f. = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 20*x^5 + 41*x^6 + 85*x^7 + 178*x^8 + ...

Mathematica

a[ n_] := (ChebyshevT[n + 1, (1 + Sqrt[17])/4] - ChebyshevT[n + 1, (1 - Sqrt[17])/4]) 2 / Sqrt[17] // Simplify; (* Michael Somos, Dec 20 2016 *)

Prog

(PARI) {a(n) = my(A); n = abs(n+1)-1; if( n<2, n>=0, n++; A = vector(n, i, 1); for(i=3, n, A[i] = A[i-1] + A[i-2]*I); norm(A[n]))}; /* Michael Somos, Apr 28 2005 */
(PARI) {a(n)=norm(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)), n))} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff((1-x^2)/(1-x-2*x^2-x^3+x^4)+x*O(x^n), n)}

Crossrefs

Cf. A092886, A201837, A201838.

Cf. A006131, A100047, A240513

Sequence in context: A360881 A030137 A243080 * A192572 A300531 A097163

Adjacent sequences: A105306 A105307 A105308 * A105310 A105311 A105312

Keyword

nonn,easy

Author

Gerald McGarvey, Apr 25 2005

Status

approved