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A001922
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Numbers n such that 3*n^2-3*n+1 is both a square (A000290) and a centered hexagonal number (A003215).
(Formerly M4569 N1946)
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7
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1, 8, 105, 1456, 20273, 282360, 3932761, 54776288, 762935265, 10626317416, 148005508553, 2061450802320, 28712305723921, 399910829332568, 5570039304932025, 77580639439715776, 1080558912851088833, 15050244140475527880, 209622859053806301481
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OFFSET
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0,2
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COMMENTS
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Also smaller of two consecutive integers whose cubes differ by a square. Defined by (a(n)+1)^3 - a(n)^3 = square.
Let m be the n-th ratio 2/1, 7/4, 26/15, 97/56, 362/209, ... Then a(n)=m*(2-m)/(m^2-3). The numerators 2, 7, 26, ... of m are A001075. The denominators 1, 4, 15, ... of m are A001353.
From Colin Barker, Jan 06 2015: (Start)
Also indices of centered triangular numbers (A005448) which are also centered square numbers (A001844).
Also indices of centered hexagonal numbers (A003215) which are also centered octagonal numbers (A016754).
Also positive integers x in the solutions to 3*x^2-4*y^2-3*x+4*y = 0, the corresponding values of y being A156712.
(End)
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..800
J. Brenner and E. P. Starke, Problem E702, Amer. Math. Monthly, 53 (1946), 465.
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.
Sociedad Magic Penny Patagonia, Leonardo en Patagonia
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
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FORMULA
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a(n) = 15a(n-1) - 15a(n-2) + a(n-3).
a(n) = (s1*t1^n + s2*t2^n + 6)/12 where s1=3+2*sqrt(3), s2=3-2*sqrt(3), t1=7+4*sqrt(3), t2=7-4*sqrt(3).
a(n) = A001075(n)*A001353(n+1).
G.f.: (1-7*x)/(1-15*x+15*x^2-x^3). - Simon Plouffe (see MAPLE line) and Colin Barker, Jan 01 2012
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EXAMPLE
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8 is in the sequence because 3*8^2-3*8+1 = 169 is a square and also a centered hexagonal number. - Colin Barker, Jan 07 2015
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MAPLE
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A001922:=(-1+7*z)/(z-1)/(z**2-14*z+1); [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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With[{s1=3+2Sqrt[3], s2=3-2Sqrt[3], t1=7+4Sqrt[3], t2=7-4Sqrt[3]}, Simplify[ Table[(s1 t1^n+s2 t2^n+6)/12, {n, 0, 20}]]] (* or *) LinearRecurrence[ {15, -15, 1}, {1, 8, 105}, 21] (* Harvey P. Dale, Aug 14 2011 *)
CoefficientList[Series[(1-7*x)/(1-15*x+15*x^2-x^3), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 16 2012 *)
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PROG
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(MAGMA) I:=[1, 8, 105]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Apr 16 2012
(PARI) Vec((1-7*x)/(1-15*x+15*x^2-x^3) + O(x^100)) \\ Colin Barker, Jan 06 2015
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CROSSREFS
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Cf. A001921, A001570, A006051.
Cf. A001844, A003215, A005448, A156712, A016754.
Sequence in context: A119934 A247537 A239400 * A222839 A113551 A082735
Adjacent sequences: A001919 A001920 A001921 * A001923 A001924 A001925
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers, Jul 04 2000
Additional comments from James R. Buddenhagen, Mar 04 2001
Name improved by Colin Barker, Jan 07 2015
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STATUS
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approved
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