1,3

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

The sequence defined by {a(n) - 1} plays a role for the computation of A065414, A146486, A146487, and A146488 equivalent to the role of A001610 for A005596, A146482, A146483 and A146484, see the variable a_{2,n} in arXiv:0903.2514. - R. J. Mathar, Mar 28 2009

Except for n = 2, a(n) is the number of digits in n-th term of A049064. This can be derived form the T. Sillke link below. - Jianing Song, Apr 28 2019

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).

Indranil Ghosh, Table of n, a(n) for n = 1..6012 (terms 1..500 from T. D. Noe)

Ignacio Cascudo, On squares of cyclic codes, arXiv:1703.01267 [cs.IT], 2017. See Theorem 6.1/Table 1.

Johann Cigler, Some remarks on generalized Fibonacci and Lucas polynomials, arXiv:1912.06651 [math.CO], 2019.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.

Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.

Daniel C. Fielder, Errata:Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.

Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 91.

Matthew Macauley, Jon McCammond, Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.

M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217 (see Eq. 29).

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.

Souvik Roy, Nazim Fatès, and Sukanta Das, Reversibility of Elementary Cellular Automata with fully asynchronous updating: an analysis of the rules with partial recurrence, hal-04456320 [nlin.CG], [cs], 2024. See p. 19.

T. Sillke, The binary form of Conway's sequence

Z. Skupien, Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles, Discr. Math., 309 (2009), 6382-6390. - N. J. A. Sloane, Feb 12 2010

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

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

a(n) = trace of successive powers of matrix ({{0,0,1},{1,0,0},{0,1,1}})^n. - Artur Jasinski, Jan 10 2007

a(n) = A000930(n) + 3*A000930(n-2). - R. J. Mathar, Nov 16 2007

Logarithmic derivative of Narayana's cows sequence A000930. - Paul D. Hanna, Oct 28 2012

a(n) = w1^n + w2^n + w3^n, where w1,w2,w3 are the roots of the cubic: (-1 - x^2 + x^3), see A092526. - Gerry Martens, Jun 27 2015

G.f. = x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 21*x^8 + ...

A001609:=-(1+3*z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation

f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-3), a(1)=1, a(2)=1, a(3)=4}, a(n), remember):

map(f, [$1..100]); # Robert Israel, Jun 29 2015

Table[Tr[MatrixPower[{{0, 0, 1}, {1, 0, 0}, {0, 1, 1}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)

Table[ HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -(n/3)}, {1/2 - n/2, 1 - n/2}, -(27/4)], {n, 20}] (* Alexander R. Povolotsky, Nov 21 2008 *)

a[1] = a[2] = 1; a[3] = 4; m = 3; a[n_] := 1 + n*Sum [Binomial [n - 1 - (m - 1)*i, i - 1]/i, {i, n/m}] A001609 = Table[a[n], {n, 100}] (* Zak Seidov, Nov 21 2008 *)

LinearRecurrence[{1, 0, 1}, {1, 1, 4}, 50] (* Vincenzo Librandi, Jun 28 2015 *)

(PARI) {a(n) = if( n<1, n=-n; polcoeff( (3 + x^2) / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( x * (1 + 3*x^2) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Aug 15 2016 */

(Magma) I:=[1, 1, 4]; [n le 3 select I[n] else Self(n-1)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Jun 28 2015

Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A000930, A218439.

Cf. also A049064, A049194.

Sequence in context: A271146 A347806 A079257 * A101590 A057916 A249853

Adjacent sequences: A001606 A001607 A001608 * A001610 A001611 A001612

nonn,easy

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

More terms from Michael Somos, Oct 03 2002

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

approved