0,2

Consider the equation core(x)=core(2x+1) where core(x) is the smallest number such that x*core(x) is a square: solutions are given by a(n)^2, n>0. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 06 2002

Terms >0 give numbers k which are solutions to the inequality |round(sqrt(2)*k)/k-sqrt(2)|<1/2/sqrt(2)/k^2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 06 2006

Also numbers n such that A125650[ 6*n^2 ] is an even perfect square, where A124650(n) is a numerator of n(n+3)/(4(n+1)(n+2)) = Sum[ 1/(k(k+1)(k+2)), {k,1,n} ]. Sequence of numbers 6*n^2 is a bisection of A125651(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 30 2006

The upper principal convergents to 2^(1/2), beginning with 3/2, 17/12, 99/70, 577/408, comprise a strictly decreasing sequence; essentially, numerators=A001541 and denominators=A001542. - Clark Kimberling (ck6(AT)evansville.edu), Aug 26 2008

Even Pell numbers. [From Omar E. Pol (info(AT)polprimos.com), Dec 10 2008]

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

I. Adler, Three diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.

H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.

D. H. Lehmer, On the multiple solutions of the Pell equation, Annals Math., 30 (1928), 66-72.

Problem E1976, Amer. Math. Monthly, 75 (1968), 683-684.

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, p. 77-79.

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002; p. 480-481.

Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers of odd index, Integers, 9 (2009), 53-64.

T. D. Noe, Table of n, a(n) for n=0..100

Index entries for sequences related to linear recurrences with constant coefficients

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths

a(n)=[(3+2sqrt(2))^n-(3-2sqrt(2))^n]/2sqrt(2). G.f.: 2*x/(1-6x+x^2).

a(n) = (C^(2n) - C^(-2n))/sqrt(8) where C = sqrt(2) + 1. - Gary Adamson, May 11, 2003.

For all terms x of the sequence, 2*x^2 + 1 is a square. Lim. as n -> Inf. a(n)/a(n-1) = 3 + 2*Sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002

For n > 0: a(n) = A001652(n) + A046090(n) - A001653(n); e.g. 70 = 119 + 120 - 169. Also a(n) = A001652(n - 1) + A046090(n - 1) + A001653(n - 1); e.g. 70 = 20 + 21 + 29. Also a(n)^2 + 1 = A001653(n - 1)*A001653(n); e.g. 12^2 + 1 = 145 = 5*29. Also a(n + 1)^2 = A084703(n + 1) = A001652(n)*A001652(n + 1) + A046090(n)*A046090(n + 1). - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003

a(n) = ((1+sqrt(2))^(2*n)-(1-sqrt(2))^(2*n))/(2*sqrt(2)) - Antonio Alberto Olivares (tonioolivares(AT)todito.com), Dec 24 2003

n such that Mod(sigma(2*n^2+1), 2 ) = 1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

2*A001541(k)*A001653(n)*A001653(n+k)=A001653(n)^2+A0001653(n+k)^2+a2(k)^2; e.g., 2*3*5*29=5^2+29^2+2^2; 2*99*29*5741=2*99*29*5741=29^2+5741^2+70^2 - Charlie Marion (charliemath(AT)optonline.net), Oct 12 2007

a(n) = Sinh[2n*ArcSinh[1]]/Sqrt[2] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008

A001542:=2*z/(1-6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

(PARI) for (i=0, 10000, if(Mod(sigma(2*i^2+1), 2)==1, print1(i, ", ")))

Bisection of A000129. Cf. A001541, A007913, A003499. Equals twice A001109.

A001542(n) = sqrt{2*(A001541(n))^2-2}/2 - Barry E. Williams, May 07 2000

Cf. A125650, A125651, A125652.

Adjacent sequences: A001539 A001540 A001541 this_sequence A001543 A001544 A001545

Sequence in context: A078839 A026306 A116398 this_sequence A059229 A001251 A143357

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).