Search: id:A001542
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%I A001542 M2030 N0802
%S A001542 0,2,12,70,408,2378,13860,80782,470832,2744210,15994428,93222358,
%T A001542 543339720,3166815962,18457556052,107578520350,627013566048,
%U A001542 3654502875938,21300003689580,124145519261542,723573111879672
%N A001542 a(n) = 6a(n-1) - a(n-2).
%C A001542 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
%C A001542 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
%C A001542 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
%C A001542 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
%C A001542 Even Pell numbers. [From Omar E. Pol (info(AT)polprimos.com), Dec 10 
               2008]
%D A001542 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001542 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001542 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.
%D A001542 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 
               181-193.
%D A001542 H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles 
               Correspondance Math\'{e}matique, 4 (1878), 161-169.
%D A001542 D. H. Lehmer, On the multiple solutions of the Pell equation, Annals 
               Math., 30 (1928), 66-72.
%D A001542 Problem E1976, Amer. Math. Monthly, 75 (1968), 683-684.
%D A001542 Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, 
               p. 77-79.
%D A001542 Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and 
               Number, World Scientific, 2002; p. 480-481.
%D A001542 Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers 
               of odd index, Integers, 9 (2009), 53-64.
%H A001542 T. D. Noe, <a href="b001542.txt">Table of n, a(n) for n=0..100</a>
%H A001542 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001542 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001542 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001542 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001542 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">
               Moments, Narayana numbers and the cut and paste for lattice paths</
               a>
%F A001542 a(n)=[(3+2sqrt(2))^n-(3-2sqrt(2))^n]/2sqrt(2). G.f.: 2*x/(1-6x+x^2).
%F A001542 a(n) = (C^(2n) - C^(-2n))/sqrt(8) where C = sqrt(2) + 1. - Gary Adamson, 
               May 11, 2003.
%F A001542 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
%F A001542 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
%F A001542 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
%F A001542 n such that Mod(sigma(2*n^2+1), 2 ) = 1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), 
               Mar 26 2004
%F A001542 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
%F A001542 a(n) = Sinh[2n*ArcSinh[1]]/Sqrt[2] - Herbert Kociemba (kociemba(AT)t-online.de), 
               Apr 24 2008
%p A001542 A001542:=2*z/(1-6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
%o A001542 (PARI) for (i=0,10000,if(Mod(sigma(2*i^2+1),2)==1,print1(i,",")))
%Y A001542 Bisection of A000129. Cf. A001541, A007913, A003499. Equals twice A001109.
%Y A001542 A001542(n) = sqrt{2*(A001541(n))^2-2}/2 - Barry E. Williams, May 07 2000
%Y A001542 Cf. A125650, A125651, A125652.
%Y A001542 Sequence in context: A078839 A026306 A116398 this_sequence A059229 A001251 
               A143357
%Y A001542 Adjacent sequences: A001539 A001540 A001541 this_sequence A001543 A001544 
               A001545
%K A001542 nonn,easy,nice
%O A001542 0,2
%A A001542 N. J. A. Sloane (njas(AT)research.att.com).

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