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A143699
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a(n) = 19*a(n-1) - 41*a(n-2) + 19*a(n-3) - a(n-4).
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6
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0, 1, 19, 319, 5301, 88000, 1460701, 24245719, 402446619, 6680076601, 110880352000, 1840465787401, 30549274537419, 507077165538919, 8416803858813901, 139707705280792000, 2318961358994380101
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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)
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MATHEMATICA
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LinearRecurrence[{19, -41, 19, -1}, {0, 1, 19, 319}, 20] (* Jean-François Alcover, Dec 12 2016 *)
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PROG
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(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
(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
(Sage)
def A143699_list(prec):
P.<x> = 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
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CROSSREFS
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Cf. A003733, A100047.
Sequence in context: A166965 A137352 A027541 * A015676 A098304 A014900
Adjacent sequences: A143696 A143697 A143698 * A143700 A143701 A143702
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, based on email from R. K. Guy, Feb 08 2009
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STATUS
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approved
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