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A006077 (n+1)^2*a(n+1) = (9n^2+9n+3)*a(n) - 27*n^2*a(n-1), with a(0) = 1 and a(1) = 3.
(Formerly M2775)
4
1, 3, 9, 21, 9, -297, -2421, -12933, -52407, -145293, -35091, 2954097, 25228971, 142080669, 602217261, 1724917221, 283305033, -38852066421, -337425235479, -1938308236731, -8364863310291, -24286959061533, -3011589296289, 574023003011199, 5028616107443691 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004

Conjecture: Let W(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,n. If n == 1 (mod 3) then W(n) = 0. When n == 0 or 2 (mod 3), W(n)*(-1)^(floor[(n+1)/3])/6^n is always a positive odd integer. - Zhi-Wei Sun, Aug 21 2013

Conjecture: Let p == 1 (mod 3) be a prime, and write 4*p = x^2 + 27*y^2 with x, y integers and x == 1 (mod 3). Then W(p-1) == (-1)^{(p+1)/2}*(x-p/x) (mod p^2), where W(n) is defined as the above. - Zhi-Wei Sun, Aug 23 2013

REFERENCES

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Z.-W. Sun, Connnections between p = x^2+3*y^2 and Franel numbers, J. Number Theory 133(2013), 2914-2928.

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Zhi-Wei Sun, Three mysterious conjectures on Hankel-type determinants, a message to Number Theory List, August 22, 2013.

FORMULA

G.f.: hypergeom([1/3, 2/3],[1],x^3/(x-1/3)^3)/(1-3*x). - Mark van Hoeij, Oct 25 2011

It is known that a(n) = sum_{k=0}^{[n/3]}(-1)^k*3^(n-3k)*C(n,3k)*C(2k,k)*C(3k,k). - Zhi-Wei Sun, Aug 21 2013

PROG

(PARI) subst(eta(q)^3/eta(q^3), q, serreverse(eta(q^9)^3/eta(q)^3*q)) (generating function) \\ Helena Verrill (verrill(AT)math.lsu.edu), Apr 20 2009

CROSSREFS

Sequence in context: A196212 A146219 A197403 * A109612 A032668 A050839

Adjacent sequences:  A006074 A006075 A006076 * A006078 A006079 A006080

KEYWORD

sign

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000

STATUS

approved

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Last modified July 6 06:44 EDT 2015. Contains 259270 sequences.