Search: id:A001921
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%I A001921 M4455 N1885
%S A001921 0,7,104,1455,20272,282359,3932760,54776287,762935264,10626317415,
%T A001921 148005508552,2061450802319,28712305723920,399910829332567,
%U A001921 5570039304932024,77580639439715775,1080558912851088832
%N A001921 a(n) = 14a(n-1) - a(n-2) + 6.
%C A001921 (a(n)+1)^3 - a(n)^3 is a square (that of A001570(n)).
%C A001921 Define a(1)=0 a(2)=7 such that 3*(a(1)^2)+3*a(1)+1=j(1)^2=1^2 and 3*(a(2)^2)+3*a(2)+1=j(2)^2=13^2. 
               Then a(n)=a(n-2)+8*sqrt(3*(a(n-1)^2)+3*a(n-1)+1). Another definition 
               : a(n) such that 3*(a(n)^2)+3*a(n)+1 = j(n)^2. - Pierre CAMI (pierrecami(AT)tele2.fr), 
               Mar 30 2005
%C A001921 a(n)=A001353(n)*A001075(n+1). For n>0, the triple {a(n), a(n)+1=A001922(n), 
               A001570(n)} forms a near-isoceles triangle with angle 2*pi/3 bounded 
               by the consecutive sides. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jul 21 2006
%D A001921 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001921 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001921 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 A001921 Problem E702, Amer. Math. Monthly, 53 (1946), 465.
%D A001921 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 
               104.
%H A001921 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 A001921 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 A001921 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexNumber.html">Hex Number</a>
%F A001921 The ratio A001570(n)/A001921(n) tends to sqrt(3) ( 1.73205...) as n increases. 
               - Pierre CAMI (pierrecami(AT)tele2.fr), Apr 21 2005
%F A001921 a(n)=-1/2-(1/6)*sqrt(3)*[7-4*sqrt(3)]^n+(1/6)*sqrt(3)*[7+4*sqrt(3)]^n+(1/
               4)*[7+4*sqrt(3)]^n +(1/4)*[7-4*sqrt(3)]^n, with n>=0 - Paolo P. Lava 
               (ppl(AT)spl.at), Jun 19 2008
%F A001921 a(n)=(A028230(n+1)-1)/2. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 19 2009]
%p A001921 A001921:=z*(-7+z)/(z-1)/(z**2-14*z+1); [Conjectured by S. Plouffe in 
               his 1992 dissertation.]
%Y A001921 Cf. A001922, A001570.
%Y A001921 Cf. A001570.
%Y A001921 Sequence in context: A142400 A032460 A101746 this_sequence A098362 A093741 
               A139742
%Y A001921 Adjacent sequences: A001918 A001919 A001920 this_sequence A001922 A001923 
               A001924
%K A001921 nonn,easy
%O A001921 0,2
%A A001921 N. J. A. Sloane (njas(AT)research.att.com).
%E A001921 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 04 2000

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