OFFSET
1,1
COMMENTS
The Pell equation x^2 - a(n)*y^2 = +4 has infinitely many (integer) solutions (see A077428 and A078355).
First differences: 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 8, ... , only 4's and 8's?. - Paul Curtz, Apr 11 2019
Yes. There are only 4's and 8's. Proof: Only multiples of 4 may appear. The 4's correspond to successive composite in A016813, whereas an 8 corresponds to a square. A greater multiple of 4 would imply to have at least 2 consecutive squares in A016813, which is not possible since 2 consecutive squares cannot have a difference of 4. That sequence of 4's and 8's can be obtained with A010052 (without the 1st term) where the 0's are replaced with 4's and 1's replaced with 8's. - Michel Marcus, Apr 16 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. R. Finch, Class number theory [Cached copy, with permission of the author]
A. M. Legendre, Expression les plus simples des formules Ly^2+Myz+Nz^2 où M est impair pour toutes les valeurs de B = M^2-4LN depuis B=5 jusqu'à B=305, Essai sur la Théorie des Nombres An VI, Table II. [Paul Curtz, Apr 11 2019]
MAPLE
A077425 := proc(n::integer) local resul, i ; resul := 5 ; i := 1 ; while i < n do resul := resul+4 ; while issqr(resul) do resul := resul+4 ; od ; i:= i+1 ; od ; RETURN(resul) ; end proc:
seq(A077425(n), n=1..31) ; # R. J. Mathar, Apr 25 2006
MATHEMATICA
Select[Range[5, 300, 4], !IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Dec 05 2012 *)
PROG
(PARI) [n | n <- vector(100, n, 4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014
(PARI) list(lim)=my(v=List()); for(s=2, sqrtint((lim\=1)+1), forstep(n=s^2 + if(s%2, 4, 1), min((s+1)^2-1, lim), 4, listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Nov 04 2021
(Python)
from operator import sub
from sympy import integer_nthroot
def A077425(n): return n+sub(*integer_nthroot(n, 2))<<2|1 # Chai Wah Wu, Oct 01 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
EXTENSIONS
More terms from Max Alekseyev, Mar 03 2010
STATUS
approved