A001638 A Fielder sequence: a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4. (Formerly M3351 N1348)

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

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; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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

Wikipedia, Companion matrix.

A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no.

6, 2006, pp. 840-855.

Index to sequences related to Olympiads.

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 A000032.

a(n) = a(n-1) + a(n-3) + a(n-4). - Eric W. Weisstein, Apr 10 2018

a(n) = Trace(M^n), where M = [0, 0, 0, 1; 1, 0, 0, 1; 0, 1, 0, 0; 0, 0, 1, 1] is the companion matrix to the monic polynomial x^4 - x^3 - x - 1. . It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Jan 08 2023

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

Cf. A000032, A001609, A001634 - A001636, A001638 - A001645, A001648, A001649.

Sequence in context: A209571 A269845 A124258 * A133826 A209565 A122185

Adjacent sequences: A001635 A001636 A001637 * A001639 A001640 A001641

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Feb 17 2002 and Nov 02 2002

Status

approved