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A006452
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a(n)=6a(n-2)-a(n-4).
(Formerly M1245)
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7
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1, 1, 2, 4, 11, 23, 64, 134, 373, 781, 2174, 4552, 12671, 26531, 73852, 154634, 430441, 901273, 2508794, 5253004, 14622323, 30616751, 85225144, 178447502, 496728541, 1040068261, 2895146102, 6061962064, 16874148071, 35331704123
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Solution to a Diophantine equation.
n such that n^2-1 is a triangular number - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002
For all elements "x" of the sequence, 8*x^2 - 7 is a square. Lim n-> inf. a(n)/a(n-2) = 3 + Sqrt(8). If n is odd, Lim n -> inf. a(n)/a(n-1) = (9 + 2*Sqrt(8))/7. If n is even, Lim n -> inf. a(n)/a(n-1) = (11 + 3*Sqrt(8))/7. - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 07 2002
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REFERENCES
| A. J. Gottlieb, How four dogs meet in a field, etc., Technology Review, Jul/Aug 1973 pp. 73-74.
J. O. Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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.
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FORMULA
| a(2n) = A038725(n). a(2n+1) = A038723(n).
For n (even), a(n) = [ [(3 + Sqrt(8))^((n/2)+1) - (3 - Sqrt(8))^((n/2)+1)] - 2*[(3 + Sqrt(8))^((n/2)-1) - (3 - Sqrt(8))^((n/2)-1)] ] / (6*Sqrt(8)) For n (odd), a(n) = [ [(3 + Sqrt(8))^((n+1)/2) - (3 - Sqrt(8))^((n+1)/2)] - 2*[(3 + Sqrt(8))^((n-1)/2) - (3 - Sqrt(8))^((n-1)/2)] ] / (2*Sqrt(8)) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 07 2002
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MAPLE
| A006452:=-(z-1)*(z**2+3*z+1)/(z**2+2*z-1)/(z**2-2*z-1); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for one of the leading 1's.]
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MATHEMATICA
| s=0; lst={1}; Do[s+=n; If[Sqrt[s+1]==Floor[Sqrt[s+1]], AppendTo[lst, Sqrt[s+1]]], {n, 0, 8!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]
a[0] = a[1] = 1; a[2] = 2; a[3] = 4; a[n_] := 6 a[n - 2] - a[n - 4]; Array[a, 30, 0] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 11 2010]
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CROSSREFS
| Cf. A006451, A006454.
Sequence in context: A026531 A038047 A061152 * A104430 A103669 A034485
Adjacent sequences: A006449 A006450 A006451 * A006453 A006454 A006455
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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
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 03 2000
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