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Search: id:A077998
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| A077998 |
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Expansion of (1-x)/(1-2*x-x^2+x^3). |
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8
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| 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209, 19181424995, 43100270734, 96845429254
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Let u(k), v(k), w(k) be be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (A006054 with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = this sequence with an extra initial 0. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary Adamson, Dec 23 2003.
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A077998 counts closed walks of length n at the vertex of degree 4. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
a(n)=number of Motzkin (n+2)-sequences with no flatsteps at ground level and whose height is <=2. For example, a(3)=6 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD, UUFDD. - David Callan (callan(AT)stat.wisc.edu), Dec 09 2004
Number of compositions of n if there are two kinds of part 2. Example: a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1). Row sums of A105477. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
Diagonal sums of A056242. - Paul Barry (pbarry(AT)wit.ie), Dec 26 2007
Diagonal sums of triangle in A105306. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2008]
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REFERENCES
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Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
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FORMULA
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a(0)=a(1)=1, a(2)=3, a(n+1)=2*a(n)+a(n-1)-a(n-2) for n>=2. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 07 2006
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MATHEMATICA
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CoefficientList[Series[(1 - x)/(1 - 2*x - x^2 + x^3), {x, 0, 200}], x] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Sep 11 2006
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CROSSREFS
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Apart from initial term, same as A006356, which is the main entry for this sequence. A106803 is yet another version.
Cf. A105477.
Sequence in context: A063119 A106803 A006356 this_sequence A090165 A129954 A114945
Adjacent sequences: A077995 A077996 A077997 this_sequence A077999 A078000 A078001
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 08 2008 at the suggestion of R. J. Mathar
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