OFFSET
1,2
COMMENTS
The ansatz 3*a(n)+1=A^2, 7*a(n)+1=B^2 is equivalent to the Pell equation x^2-21*y^2=1 (see A077232 for d=21), with x=(21*a(n)+5)/2 and y=A*B/2.
Bisection of A089927. - R. J. Mathar, Jul 10 2009
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..736
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).
FORMULA
a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3).
a(n) = ((5+w)/2*((23+5*w)/2)^(n-1) + (5-w)/2*((23-5*w)/2)^(n-1) - 5)/21; where w=sqrt(21). [Corrected by Kevin Ryde, Sep 11 2020]
G.f.: 5*x^2/((1-x)*(x^2-23*x+1)). - R. J. Mathar, Jul 10 2009
From Francesca Arici, Sep 12 2020: (Start)
a(n) = 23*a(n-1) - a(n-2) + 5.
a(n) = 5*A334673(n-1). - Hugo Pfoertner, Apr 07 2021
MAPLE
j:=0: for n from 0 to 1000000 do a:=sqrt(3*n+1): b:=sqrt(7*n+1):
if (trunc(a)=a) and (trunc(b)=b) then j:=j+1: print(j, n, a, b): end if:
end do:
MATHEMATICA
LinearRecurrence[{24, -24, 1}, {0, 5, 120}, 30] (* Harvey P. Dale, Dec 17 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, May 24 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Jul 10 2009
Name edited by Michel Marcus, Sep 12 2020
STATUS
approved