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A001638
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A Fielder sequence: a(n)=a(n-1)+a(n-3)+a(n-4), n>=4.
(Formerly M3351 N1348)
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8
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4, 1, 1, 4, 9, 11, 16, 29, 49, 76, 121, 199, 324, 521, 841, 1364, 2209, 3571, 5776, 9349, 15129, 24476, 39601, 64079, 103684, 167761, 271441, 439204, 710649, 1149851, 1860496, 3010349, 4870849, 7881196, 12752041, 20633239, 33385284, 54018521
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| For n > 1, a(n) is the number of ways of choosing a subset of vertices of an n-cycle so that every vertex of the n-cycle is adjacent to one of the chosen vertices. (Note that this is not the same as the number of dominating sets of the n-cycle, which is given by A001644.) [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), Sep 12 2010]
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REFERENCES
| Fielder, Daniel C.; Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.
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
| 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.
Middle European Math Olympiad 2010, Team Problem 3. Available online at the Art of Problem Solving. [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), Sep 12 2010]
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FORMULA
| G.f.: (1-x)(4+x+x^2)/((1+x^2)(1-x-x^2)). a(n)=L(n)+i^n+(-i)^n, a(2n)=L(n)^2, a(2n+1)=L(2n+1) where L() is Lucas sequence.
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MAPLE
| A001638:=-(z+1)*(4*z**2-z+1)/(z**2+z-1)/(z**2+1); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for the initial 4.]
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PROG
| (PARI) a(n)=if(n<0, 0, fibonacci(n+1)+fibonacci(n-1)+simplify(I^n+(-I)^n))
(PARI) a(n)=if(n<0, 0, polsym((1+x-x^2)*(1+x^2), n)[n+1])
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CROSSREFS
| Sequence in context: A026998 A080061 A124258 * A133826 A122185 A193752
Adjacent sequences: A001635 A001636 A001637 * A001639 A001640 A001641
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Michael Somos, Feb 17 2002 and Nov 2 2002
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