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A001609
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a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).
(Formerly M3240 N1308)
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18
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1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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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
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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.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
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
Yüksel Soykan, On Generalized Narayana Numbers, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 7, No. 3, 43-56.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).
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FORMULA
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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). - Gerry Martens, Jun 27 2015
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EXAMPLE
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G.f. = x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 21*x^8 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A000930, A218439.
Cf. also A049064, A049194.
Sequence in context: A310570 A271146 A079257 * A101590 A057916 A249853
Adjacent sequences: A001606 A001607 A001608 * A001610 A001611 A001612
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
More terms from Michael Somos, Oct 03 2002
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STATUS
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approved
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