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]

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.

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

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

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.

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Tanya Khovanova, Recursive Sequences

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.

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

Index entries for sequences related to linear recurrences with constant coefficients

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.

Sequence in context: A078839 A026306 A116398 this_sequence A059229 A001251 A143357

Adjacent sequences: A001539 A001540 A001541 this_sequence A001543 A001544 A001545

nonn,easy,nice

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