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Search: id:A001922
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| A001922 |
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3*n^2-3*n+1 is a square hex number.
(Formerly M4569 N1946)
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+0
4
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| 1, 8, 105, 1456, 20273, 282360, 3932761, 54776288, 762935265, 10626317416, 148005508553, 2061450802320, 28712305723921, 399910829332568, 5570039304932025, 77580639439715776, 1080558912851088833
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Also smaller of two consecutive integers whose cubes differ by a square. Defined by (a(n)+1)^3 - a(n)^3 = square.
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REFERENCES
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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.
Problem E702, Amer. Math. Monthly, 53 (1946), 465.
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LINKS
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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.
Sociedad Magic Penny Patagonia, Leonardo en Patagonia
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FORMULA
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a(n) = 15a(n-1) - 15a(n-2) + a(n-3).
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).
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MAPLE
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A001922:=(-1+7*z)/(z-1)/(z**2-14*z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Cf. A001921, A001570, A006051.
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.
a(n)=A001075(n)*A001353(n+1).
Sequence in context: A141383 A034300 A119934 this_sequence A113551 A082735 A024358
Adjacent sequences: A001919 A001920 A001921 this_sequence A001923 A001924 A001925
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KEYWORD
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nonn,easy
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AUTHOR
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njas
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 04 2000
Additional comments from Jim Buddenhagen (jbuddenh(AT)gmail.com), Mar 04 2001
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