OFFSET
1,1
COMMENTS
It appears that the 4-tuple (3, (u^2-1)/3, (floor((3*n+11)/2)^2-1)/3, a(n)) has the Diophantus' property that the product of any two distinct terms plus one is a square.
LINKS
Lenny Jones, A polynomial Approach to a Diophantine Problem, Math. Mag. 72 (1999) 52-55.
Eric Weisstein's World of Mathematics, Diophantus Property.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).
FORMULA
From R. J. Mathar, May 15 2009: (Start)
a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9).
G.f.: -8*x*(126+134*x+41*x^2-65*x^3+95*x^4+52*x^5-61*x^6-13*x^7+15*x^8)/((1+x)^4*(x-1)^5). (End)
EXAMPLE
For n=1 we get the 4-tuple (3,5,16,1008), and 3*5+1=16=4^2, 3*16+1=49=7^2, 3*1008+1=3025=55^2, 5*16+1=81=9^2, 5*1008+1=5041=71^2, 16*1008+1=16129=127^2.
MATHEMATICA
Table[u=Floor[(3n+5)/2]; 4/3 u(u^3+6u^2+8u-3), {n, 30}] (* or *) LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {1008, 2080, 6440, 10208, 22360, 31416, 57408, 75208, 122816}, 30] (* Harvey P. Dale, Nov 19 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John W. Layman, May 14 2009
STATUS
approved