1,2

Equivalently, triangular hex numbers.

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.

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 19.

S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, submitted.

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

Comments from Richard Choulet (richardchoulet(AT)yahoo.fr), 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 (schlicks(AT)gvsu.edu), Apr 24 2007

a(n) = (A007667(n+1)-1)/4. - Ralf Stephan, Mar 03 2004

a(1)=91 because 91 is the sixth centered hexagonal number and the seventh hexagonal number.

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 (schlicks(AT)gvsu.edu), Apr 24 2007

A006244:=-(1-8*z+z**2)/(z-1)/(z**2-98*z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A001570, A001921, A000384, A003215.

Sequence in context: A103855 A022253 A060078 this_sequence A054216 A109627 A095372

Adjacent sequences: A006241 A006242 A006243 this_sequence A006245 A006246 A006247

nonn,easy,more

njas, Jeffrey Shallit

Edited by njas, Sep 25 2007