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A001921
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a(n) = 14a(n-1) - a(n-2) + 6.
(Formerly M4455 N1885)
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12
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0, 7, 104, 1455, 20272, 282359, 3932760, 54776287, 762935264, 10626317415, 148005508552, 2061450802319, 28712305723920, 399910829332567, 5570039304932024, 77580639439715775, 1080558912851088832, 15050244140475527879, 209622859053806301480
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OFFSET
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0,2
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COMMENTS
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(a(n)+1)^3 - a(n)^3 is a square (that of A001570(n)).
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, Mar 30 2005
a(n)=A001353(n)*A001075(n+1). For n>0, the triple {a(n), a(n)+1=A001922(n), A001570(n)} forms a near-isosceles triangle with angle 2*pi/3 bounded by the consecutive sides. - Lekraj Beedassy, Jul 21 2006
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REFERENCES
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J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
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
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T. D. Noe, Table of n, a(n) for n = 0..200
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Hex Number
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FORMULA
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The ratio A001570(n)/A001921(n) tends to sqrt(3) ( 1.73205...) as n increases. - Pierre CAMI, Apr 21 2005
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, Jun 19 2008
a(n) = (A028230(n+1)-1)/2. [From R. J. Mathar, Mar 19 2009]
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MAPLE
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A001921:=z*(-7+z)/(z-1)/(z**2-14*z+1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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t = {0, 7}; Do[AppendTo[t, 14*t[[-1]] - t[[-2]] + 6], {20}]; t (* T. D. Noe, Aug 17 2012 *)
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CROSSREFS
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Cf. A001922, A001570.
Sequence in context: A177752 A101746 A195246 * A215552 A098362 A093741
Adjacent sequences: A001918 A001919 A001920 * A001922 A001923 A001924
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, Jul 04 2000
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STATUS
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approved
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