A143699 a(n) = 19*a(n-1) - 41*a(n-2) + 19*a(n-3) - a(n-4).

0, 1, 19, 319, 5301, 88000, 1460701, 24245719, 402446619, 6680076601, 110880352000, 1840465787401, 30549274537419, 507077165538919, 8416803858813901, 139707705280792000, 2318961358994380101

Offset: 0

Author

N. J. A. Sloane, based on email from R. K. Guy, Feb 08 2009

Keywords

• nonn
• easy

Comments

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

A003733 = 5 * (A143699)^2. - R. K. Guy, Mar 11 2010

The sequence is the case P1 = 19, P2 = 39, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..750
• Per Hakan Lundow, Enumeration of matchings in polygraphs, Section 8.1.
• 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 to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (19,-41,19,-1).

Crossrefs

Cf. A003733, A100047.

Programs

Magma

I:=[0,1,19,319]; [n le 4 select I[n] else 19*Self(n-1) -41*Self(n-2) +19*Self(n-3) -Self(n-4): n in [1..30]]; // G. C. Greubel, May 31 2021

Mathematica

LinearRecurrence[{19,-41,19,-1}, {0,1,19,319}, 20] (* Jean-François Alcover, Dec 12 2016 *)

PARI

{a(n) = n = abs(n); polcoeff( x*(1-x^2)/(1 -19*x +41*x^2 -19*x^3 +x^4) + x*O(x^n), n)} \\ Michael Somos, Feb 24 2009

Sage

def A143699_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x^2)/(1 -19*x +41*x^2 -19*x^3 +x^4) ).list()
A143699_list(30) # G. C. Greubel, May 31 2021

Formula

Equals sqrt(A003733(n)/5).

G.f.: x*(1+x)*(1-x)/(1 - 19*x + 41*x^2 - 19*x^3 + x^4). - R. J. Mathar, Feb 09 2009

a(-n) = a(n). - Michael Somos, Feb 24 2009

a(n) = (r1^n + r2^n - r3^n - r4^n) / s1 where s1 = sqrt(205), s2 = sqrt(550 + 38*s1), s3 = 36 * sqrt(5) / s2, r1 = (19 + s1 + s2) / 4, r2 = 1/r1, r3 = (19 - s1 + s3) / 4, r4 = 1/r3. - Michael Somos, Feb 12 2012

From Peter Bala, Apr 03 2014: (Start)

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = (1/4)*(19 + sqrt(205)), beta = (1/4)*(19 - sqrt(205)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n)= U(n-1, (sqrt(5) - 9)/4)*U(n-1, -(sqrt(5) + 9)/4) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -39/4; 1, 19/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)