A006244 Hexagonal numbers (A000384) which are also centered hexagonal numbers (A003215). (Formerly M5363)

1, 91, 8911, 873181, 85562821, 8384283271, 821574197731, 80505887094361, 7888755361049641, 773017519495770451, 75747828155224454551, 7422514141692500775541, 727330638057709851548461, 71270980015513872950973631, 6983828710882301839343867371, 684343942686450066382748028721

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.: h(z)=((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

Cf. A001570, A001921, A000384, A003215, A253175.

Sequence in context: A172174 A060078 A202564 * A054216 A109627 A095372

Adjacent sequences: A006241 A006242 A006243 * A006245 A006246 A006247

Keyword

nonn,easy

Author

N. J. A. Sloane, Jeffrey Shallit

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