OFFSET
0,2
COMMENTS
Numbers k such that (k^2 + k + 1)/3 is a square. - Arkadiusz Wesolowski, Feb 10 2012
Given by the numerators of the convergents to the continued fraction [1,(1,2)^i,3,(1,2)^{i-1},1]. - Jeffrey Shallit, Dec 11 2017
REFERENCES
J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
T. Nagell, Des équations indéterminées x^2 + x + 1 = y^n et x^2 + x + 1 = 3*y^n, Norsk Mat. Forenings Skrifter, Ser. I, (1921).
LINKS
Michel Marcus, Table of n, a(n) for n = 0..100
Kevin A. Broughan, An explicit bound for aliquot cycles of repdigits, #A15 INTEGERS Vol 12 (2012) p. 4.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1)
FORMULA
a(n) = sqrt(3)*((2+sqrt(3))^(2*n+1) - (2-sqrt(3))^(2*n+1))/4 - 1/2 (see Kevin A. Broughan paper). - Michel Marcus, Jul 28 2012
a(n) = 15*a(n-1)-15*a(n-2)+a(n-3). G.f.: (1+7*x-2*x^2)/((1-x)*(1-14*x+x^2)). - conjectured by Colin Barker, Apr 10 2012; these follow easily from the formula.
MAPLE
f:= gfun:-rectoproc({a(n) = 15*a(n-1)-15*a(n-2)+a(n-3), a(0)=1, a(1)=22, a(2)=313}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 12 2017
MATHEMATICA
With[{k = Sqrt@ 3}, Simplify@ Array[k ((2 + k)^(2 # + 1) - (2 - k)^(2 # + 1))/4 - 1/2 &, 19, 0]] (* Michael De Vlieger, Dec 11 2017 *)
PROG
(PARI) a(n) = {w = quadgen(12); w*((2+w)^(2*n+1) - (2-w)^(2*n+1))/4 - 1/2; } /* Michel Marcus, Jul 28 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Robert Israel, Dec 12 2017
STATUS
approved