A001641 A Fielder sequence: a(n) = a(n-1) + a(n-2) + a(n-4). (Formerly M2364 N0935)

1, 3, 4, 11, 16, 30, 50, 91, 157, 278, 485, 854, 1496, 2628, 4609, 8091, 14196, 24915, 43720, 76726, 134642, 236283, 414645, 727654, 1276941, 2240878, 3932464, 6900996, 12110401, 21252275, 37295140, 65448411, 114853952, 201554638, 353703730, 620706779

Offset

1,2

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 = 1..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.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

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 entries for linear recurrences with constant coefficients, signature (1,1,0,1).

Formula

G.f.: x*(1+2*x+4*x^3)/(1-x-x^2-x^4).

a(n) = n*Sum_{k=1..n} Sum_{j=floor((4*k-n)/3)..floor((4*k-n)/2)} binomial(j,n-4*k+3*j)*binomial(k,j)/k. - Vladimir Kruchinin, May 25 2011

a(n) = Trace(M^n), where M = [0, 0, 0, 1; 1, 0, 0, 0; 0, 1, 0, 1; 0, 0, 1, 1] is the companion matrix to the monic polynomial x^4 - x^3 - x^2 - 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, Dec 31 2022

Maple

A001641:=-(1+2*z+4*z**3)/(z+1)/(z**3-z**2+2*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{1, 1, 0, 1}, {1, 3, 4, 11}, 50] (* T. D. Noe, Aug 09 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(x*(1+2*x+4*x^3)/(1-x-x^2-x^4)+x*O(x^n), n))
(Maxima) a(n):=(sum(sum(binomial(j, n-4*k+3*j)*binomial(k, j), j, floor((4*k-n)/3), floor((4*k-n)/2))/k, k, 1, n))*n; /* Vladimir Kruchinin, May 25 2011 */
(Magma) I:=[1, 3, 4, 11]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 09 2018

Crossrefs

Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649, A060945.

Sequence in context: A266384 A393241 A248825 * A374712 A007382 A127804

Adjacent sequences: A001638 A001639 A001640 * A001642 A001643 A001644

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved