0,2

(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 (pierrecami(AT)tele2.fr), 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-isoceles triangle with angle 2*pi/3 bounded by the consecutive sides. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 21 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.

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.

Eric Weisstein's World of Mathematics, Hex Number

The ratio A001570(n)/A001921(n) tends to sqrt(3) ( 1.73205...) as n increases. - Pierre CAMI (pierrecami(AT)tele2.fr), 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 (ppl(AT)spl.at), Jun 19 2008

a(n)=(A028230(n+1)-1)/2. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 19 2009]

A001921:=z*(-7+z)/(z-1)/(z**2-14*z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A001922, A001570.

Cf. A001570.

Sequence in context: A142400 A032460 A101746 this_sequence A098362 A093741 A139742

Adjacent sequences: A001918 A001919 A001920 this_sequence A001922 A001923 A001924

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 04 2000