OFFSET
1,2
COMMENTS
Equivalently, triangular hex numbers.
REFERENCES
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..500
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
Eric Weisstein's World of Mathematics, Hex Number
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).
FORMULA
From Richard Choulet, Sep 19 2007: (Start)
We must solve 2*r^2-r=3*p^2-3*p+1, which gives X^2=6*Y^2+3 with X=4*r-1 and Y=2*p-1. We obtain at the same time the following sequences:
X is given by 3, 27, 267, ... sequence for which a(n+2)=10*a(n+1)-a(n) and a(n+1)=5*a(n)+2*(6a(n)^2-18)^0.5
Y is given by 1, 11, 109, ... sequence for which a(n+2)=10*a(n+1)-a(n) and a(n+1)=5*a(n)+2*(6a(n)^2+3)^0.5
p is given by 1, 6, 55, 540, ... sequence for which a(n+2)=10*a(n+1)-a(n)-4 and a(n+1)=5*a(n)-2+(24*a(n)^2-24*a(n)+9)^0.5
r is given by 1, 7, 67, 661, ... sequence for which a(n+2)=10*a(n+1)-a(n)-2 and a(n+1)=5*a(n)-1+(24*a(n)^2-12*a(n)-3)^0.5
a(n+2) = 98*a(n+1)-a(n)-6, a(n+1)=49*a(n)-3+5*(96*a(n)^2-12*a(n)-3)^0.5.
G.f.: z*(1-8*z+z^2)/((1-z)*(1-98*z+z^2)). (End)
Define x(n) + y(n)*sqrt(24) = (6+sqrt(24))*(5+sqrt(24))^n, s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+6*(s(n)^2-s(n))). - Steven Schlicker, Apr 24 2007
a(n) = (A007667(n+1)-1)/4. - Ralf Stephan, Mar 03 2004
a(n) = 99*a(n-1)-99*a(n-2)+a(n-3). - Colin Barker, Jan 08 2015
EXAMPLE
a(1)=91 because 91 is the sixth centered hexagonal number and the seventh hexagonal number.
MAPLE
CP := n -> 1+1/2*6*(n^2-n): N:=10: u:=5: v:=1: x:=6: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+24*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp), CP(s)]: end do: k_pcp; # Steven Schlicker, Apr 24 2007
A006244:=-(1-8*z+z**2)/(z-1)/(z**2-98*z+1); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.
a := n -> (Matrix([[91, 1, 1]]). Matrix([[99, 1, 0], [ -99, 0, 1], [1, 0, 0]])^n)[1, 3]; seq (a(n), n=1..20); # Alois P. Heinz, Aug 14 2008
MATHEMATICA
CoefficientList[Series[(1 - 8*x + x^2)/(1 - 99*x + 99*x^2 - x^3), {x, 0, 20}], x] (* Jean-François Alcover, Feb 26 2015 *)
PROG
(PARI) Vec(-x*(x^2-8*x+1)/((x-1)*(x^2-98*x+1)) + O(x^100)) \\ Colin Barker, Jan 08 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Sep 25 2007
More terms from Alois P. Heinz, Aug 14 2008
More terms from Jon E. Schoenfield, Dec 26 2008
STATUS
approved