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Search: id:A005386
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| A005386 |
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Area of n-th triple of squares around a triangle.
(Formerly M3017)
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+0
7
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| 1, 3, 16, 75, 361, 1728, 8281, 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n)*(-1)^(n+1) is the r=-3 member of the r-family of sequences S_r(n), n>=1, defined in A092184 where more information can be found.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
J. C. G. Nottrot, Vierkantenkransen rond een driehoek, Pythagoras (Netherlands), 14 (1975-1976) 77-81.
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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.
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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GF=x*(1-x)/(x^3-4*x^2-4*x+1), a(n)=4*(a(n-1)+a(n-2))-a(n-3), a(1)=1, a(2)=3, a(3)=16
a(n)=(2/7)*(T(n, 5/2)-(-1)^n) with twice Chebyshev's polynomials of the first kind evaluated at x=5/2: 2*T(n, 5/2)=A003501(n)= ((5+sqrt(21))^n + (5-sqrt(21))^n)/2^n. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
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MAPLE
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A005386:=-(-1+z)/(z+1)/(z**2-5*z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]
(Maple) a := n -> (Matrix([[0, 1, 3]]). Matrix(3, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, 4, -1][i] else 0 fi)^(n))[1, 1] ; seq (a(n), n=1..22); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 05 2008]
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MATHEMATICA
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a[n_]:=Module[{n1=1, n2=0}, Do[{n1, n2}={Sqrt[3]*n1+n2, n1}, {n-1}]; n1^2] a[n_]:=Round[((5+Sqrt[21])/2)^n/7] (CoefficientList[Series[{(x/(1-x*(Sqrt[3]+x)))}, {x, 0, 20}], x])^2 CoefficientList[Series[{x*(1-x)/(x^3-4*x^2-4*x+1)}, {x, 0, 20}], x]
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CROSSREFS
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Essentially the same as A003769. First differences of A099025.
Sequence in context: A004303 A005947 A003769 this_sequence A053572 A055842 A037773
Adjacent sequences: A005383 A005384 A005385 this_sequence A005387 A005388 A005389
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KEYWORD
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nonn
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AUTHOR
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Jean Meeus
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EXTENSIONS
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Edited by Peter J. C. Moses (mows(AT)mopar.freeserve.co.uk), Apr 23 2004
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
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