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Search: id:A006244
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| A006244 |
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Numbers that are simultaneously hexagonal and centered hexagonal. (Formerly M5363)
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+0 2
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| 1, 91, 8911, 873181, 85562821, 8384283271, 821574197731, 80505887094361, 7888755361049641, 773017519495770451, 75747828155224454551, 7422514141692500775541
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Equivalently, triangular hex numbers.
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REFERENCES
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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.
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LINKS
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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
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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
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a(1)=91 because 91 is the sixth centered hexagonal number and the seventh hexagonal number.
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MAPLE
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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.]
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CROSSREFS
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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
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KEYWORD
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nonn,easy,more
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AUTHOR
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njas, Jeffrey Shallit
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EXTENSIONS
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Edited by njas, Sep 25 2007
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