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 A001638 A Fielder sequence: a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4. (Formerly M3351 N1348) 10
 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, 87403801 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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.) - Joel B. Lewis, Sep 12 2010 For n >= 3, a(n) is also the number of total dominating sets in the n-cycle graph. - Eric W. Weisstein, Apr 10 2018 REFERENCES 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). LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70. 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. Middle European Math Olympiad 2010, Team Problem 3. Available online at the Art of Problem Solving. - Joel B. Lewis, Sep 12 2010 Eric Weisstein's World of Mathematics, Cycle Graph Eric Weisstein's World of Mathematics, Total Dominating Set Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1). 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. a(n) = a(n-1) + a(n-3) + a(n-4). - Eric W. Weisstein, Apr 10 2018 MAPLE A001638:=-(z+1)*(4*z**2-z+1)/(z**2+z-1)/(z**2+1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 4 MATHEMATICA LinearRecurrence[{1, 0, 1, 1}, {4, 1, 1, 4}, 50] (* T. D. Noe, Aug 09 2012 *) Table[LucasL[n] + 2 Cos[n Pi/2], {n, 0, 20}] (* Eric W. Weisstein, Apr 10 2018 *) CoefficientList[Series[(-4 + 3 x + x^3)/(-1 + x + x^3 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 10 2018 *) 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]) (MAGMA) I:=[4, 1, 1, 4]; [n le 4 select I[n] else Self(n-1) + Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 09 2018 CROSSREFS Sequence in context: A209571 A269845 A124258 * A133826 A209565 A122185 Adjacent sequences:  A001635 A001636 A001637 * A001639 A001640 A001641 KEYWORD nonn AUTHOR EXTENSIONS Edited by Michael Somos, Feb 17 2002 and Nov 02 2002 STATUS approved

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Last modified July 14 00:36 EDT 2020. Contains 335716 sequences. (Running on oeis4.)