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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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4
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1, 8, 105, 1456, 20273, 282360, 3932761, 54776288, 762935265, 10626317416, 148005508553, 2061450802320, 28712305723921, 399910829332568, 5570039304932025, 77580639439715776, 1080558912851088833
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
| Problem E702, Amer. Math. Monthly, 53 (1946), 465.
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).
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LINKS
| 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
| 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).
G.f.: (1-7*x)/(1-15*x+15*x^2-x^3). - S. Plouffe (see MAPLE line) and Colin Barker, Jan 01 2012
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MAPLE
| A001922:=(-1+7*z)/(z-1)/(z**2-14*z+1); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| With[{s1=3+2Sqrt[3], s2=3-2Sqrt[3], t1=7+4Sqrt[3], t2=7-4Sqrt[3]}, Simplify[ Table[(s1 t1^n+s2 t2^n+6)/12, {n, 0, 20}]]] (* or *) LinearRecurrence[ {15, -15, 1}, {1, 8, 105}, 21] (* From Harvey P. Dale, Aug 14 2011 *)
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CROSSREFS
| 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: A034300 A146346 A119934 * A113551 A082735 A024358
Adjacent sequences: A001919 A001920 A001921 * A001923 A001924 A001925
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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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