Search: id:A001922
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%I A001922 M4569 N1946
%S A001922 1,8,105,1456,20273,282360,3932761,54776288,762935265,10626317416,
%T A001922 148005508553,2061450802320,28712305723921,399910829332568,
%U A001922 5570039304932025,77580639439715776,1080558912851088833
%N A001922 3*n^2-3*n+1 is a square hex number.
%C A001922 Also smaller of two consecutive integers whose cubes differ by a square. 
               Defined by (a(n)+1)^3 - a(n)^3 = square.
%D A001922 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001922 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001922 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 A001922 Problem E702, Amer. Math. Monthly, 53 (1946), 465.
%H A001922 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 A001922 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 A001922 Sociedad Magic Penny Patagonia, <a href="http://www.magicpenny.org/engteorema.htm">
               Leonardo en Patagonia</a>
%F A001922 a(n) = 15a(n-1) - 15a(n-2) + a(n-3).
%F A001922 a(n)=(s1*t1^n + s2*t2^n + 6)/12 where s1=3+2*sqrt(3), s2=3-2*sqrt(3), 
               t1=7+4*sqrt(3), t2=7-4*sqrt(3).
%p A001922 A001922:=(-1+7*z)/(z-1)/(z**2-14*z+1); [Conjectured by S. Plouffe in 
               his 1992 dissertation.]
%Y A001922 Cf. A001921, A001570, A006051.
%Y A001922 Let m be the n-th ratio 2/1, 7/4, 26/15, 97/56, 362/209, ... Then a(n)=m*(2-m)/
               (m^2-3). The numerators 2, 7, 26, ... of m are A001075. The denominators 
               1, 4, 15, ... of m are A001353.
%Y A001922 a(n)=A001075(n)*A001353(n+1).
%Y A001922 Sequence in context: A034300 A146346 A119934 this_sequence A113551 A082735 
               A024358
%Y A001922 Adjacent sequences: A001919 A001920 A001921 this_sequence A001923 A001924 
               A001925
%K A001922 nonn,easy
%O A001922 0,2
%A A001922 N. J. A. Sloane (njas(AT)research.att.com).
%E A001922 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 04 2000
%E A001922 Additional comments from Jim Buddenhagen (jbuddenh(AT)gmail.com), Mar 
               04 2001

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