login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001921 a(n) = 14a(n-1) - a(n-2) + 6.
(Formerly M4455 N1885)
17
0, 7, 104, 1455, 20272, 282359, 3932760, 54776287, 762935264, 10626317415, 148005508552, 2061450802319, 28712305723920, 399910829332567, 5570039304932024, 77580639439715775, 1080558912851088832, 15050244140475527879, 209622859053806301480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(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

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Hex Number

FORMULA

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]

MAPLE

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

MATHEMATICA

t = {0, 7}; Do[AppendTo[t, 14*t[[-1]] - t[[-2]] + 6], {20}]; t (* T. D. Noe, Aug 17 2012 *)

CROSSREFS

Cf. A001922, A001570.

Cf. numbers m such that k*A000217(m)+1 is a square: A006451 for k=1; m=0 for k=2; A233450 for k=3; A001652 for k=4; A129556 for k=5; this sequence for k=6. [Bruno Berselli, Dec 16 2013]

Sequence in context: A101746 A195246 A224706 * A215552 A098362 A093741

Adjacent sequences:  A001918 A001919 A001920 * A001922 A001923 A001924

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Jul 04 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 24 09:23 EST 2014. Contains 249882 sequences.