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A006244
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Numbers that are simultaneously hexagonal and centered hexagonal.
(Formerly M5363)
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2
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1, 91, 8911, 873181, 85562821, 8384283271, 821574197731, 80505887094361, 7888755361049641, 773017519495770451, 75747828155224454551, 7422514141692500775541, 727330638057709851548461, 71270980015513872950973631, 6983828710882301839343867371, 684343942686450066382748028721
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Equivalently, triangular hex numbers.
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REFERENCES
| M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 19.
S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Jon E. Schoenfield, Table of n, a(n) for n=1..500 [From Jon E. Schoenfield (jonscho(AT)hiwaay.net), Dec 26 2008]
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
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FORMULA
| 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
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EXAMPLE
| a(1)=91 because 91 is the sixth centered hexagonal number and the seventh hexagonal number.
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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 (schlicks(AT)gvsu.edu), Apr 24 2007
A006244:=-(1-8*z+z**2)/(z-1)/(z**2-98*z+1); [Conjectured (correctly) by S. 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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008]
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CROSSREFS
| Cf. A001570, A001921, A000384, A003215.
Sequence in context: A172174 A060078 A202564 * A054216 A109627 A095372
Adjacent sequences: A006241 A006242 A006243 * A006245 A006246 A006247
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 25 2007
More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008
More terms from Jon E. Schoenfield (jonscho(AT)hiwaay.net), Dec 26 2008
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