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 A006244 Hexagonal numbers (A000384) which are also centered hexagonal numbers (A003215). (Formerly M5363) 3
 1, 91, 8911, 873181, 85562821, 8384283271, 821574197731, 80505887094361, 7888755361049641, 773017519495770451, 75747828155224454551, 7422514141692500775541, 727330638057709851548461, 71270980015513872950973631, 6983828710882301839343867371, 684343942686450066382748028721 (list; graph; refs; listen; history; text; internal format)
 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. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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 (schlicks(AT)gvsu.edu), 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:=: 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 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 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

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Last modified November 11 22:31 EST 2019. Contains 329046 sequences. (Running on oeis4.)