OFFSET

0,4

COMMENTS

Named after a 14th-century Indian mathematician. [The sequence first appeared in the book "Ganita Kaumudi" (1356) by the Indian mathematician Narayana Pandita (c. 1340 - c. 1400). - Amiram Eldar, Apr 15 2021]

Number of compositions of n into parts 1 and 3. - Joerg Arndt, Jun 25 2011

A Lamé sequence of higher order.

Could have begun 1,0,0,1,1,1,2,3,4,6,9,... (A078012) but that would spoil many nice properties.

Number of tilings of a 3 X n rectangle with straight trominoes.

Number of ways to arrange n-1 tatami mats in a 2 X (n-1) room such that no 4 meet at a point. For example, there are 6 ways to cover a 2 X 5 room, described by 11111, 2111, 1211, 1121, 1112, 212.

Equivalently, number of compositions (ordered partitions) of n-1 into parts 1 and 2 with no two 2's adjacent. E.g., there are 6 such ways to partition 5, namely 11111, 2111, 1211, 1121, 1112, 212, so a(6) = 6. [Minor edit by Keyang Li, Oct 10 2020]

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 = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..floor(n/m)} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.

a(n+2) is the number of n-bit 0-1 sequences that avoid both 00 and 010. - David Callan, Mar 25 2004 [This can easily be proved by the Cluster Method - see for example the Noonan-Zeilberger article. - N. J. A. Sloane, Aug 29 2013]

a(n-4) is the number of n-bit sequences that start and end with 0 but avoid both 00 and 010. For n >= 6, such a sequence necessarily starts 011 and ends 110; deleting these 6 bits is a bijection to the preceding item. - David Callan, Mar 25 2004

Also number of compositions of n+1 into parts congruent to 1 mod m. Here m=3, A003269 for m=4, etc. - Vladeta Jovovic, Feb 09 2005

Row sums of Riordan array (1/(1-x^3), x/(1-x^3)). - Paul Barry, Feb 25 2005

Row sums of Riordan array (1,x(1+x^2)). - Paul Barry, Jan 12 2006

Starting with offset 1 = row sums of triangle A145580. - Gary W. Adamson, Oct 13 2008

Number of digits in A061582. - Dmitry Kamenetsky, Jan 17 2009

From Jon Perry, Nov 15 2010: (Start)

The family a(n) = a(n-1) + a(n-m) with a(n)=1 for n=0..m-1 can be generated by considering the sums (A102547):

1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 10

1 3 6 10 15 21 28

1 4 10 20

1

------------------------------

1 1 1 2 3 4 6 9 13 19 28 41 60

with (in this case 3) leading zeros added to each row.

(End)

Number of pairs of rabbits existing at period n generated by 1 pair. All pairs become fertile after 3 periods and generate thereafter a new pair at all following periods. - Carmine Suriano, Mar 20 2011

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=3, 2*a(n-3) equals the number of 2-colored compositions of n with all parts >= 3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

For n>=2, row sums of Pascal's triangle (A007318) with triplicated diagonals. - Vladimir Shevelev, Apr 12 2012

Pisano period lengths of the sequence read mod m, m >= 1: 1, 7, 8, 14, 31, 56, 57, 28, 24, 217, 60, 56, 168, ... (A271953) If m=3, for example, the remainder sequence becomes 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, ... with a period of length 8. - R. J. Mathar, Oct 18 2012

Diagonal sums of triangle A011973. - John Molokach, Jul 06 2013

"In how many ways can a kangaroo jump through all points of the integer interval [1,n+1] starting at 1 and ending at n+1, while making hops that are restricted to {-1,1,2}? (The OGF is the rational function 1/(1 - z - z^3) corresponding to A000930.)" [Flajolet and Sedgewick, p. 373] - N. J. A. Sloane, Aug 29 2013

a(n) is the number of length n binary words in which the length of every maximal run of consecutive 0's is a multiple of 3. a(5) = 4 because we have: 00011, 10001, 11000, 11111. - Geoffrey Critzer, Jan 07 2014

a(n) is the top left entry of the n-th power of the 3X3 matrix [1, 0, 1; 1, 0, 0; 0, 1, 0] or of the 3 X 3 matrix [1, 1, 0; 0, 0, 1; 1, 0, 0]. - R. J. Mathar, Feb 03 2014

a(n-3) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 1, 0; 0, 1, 1; 1, 0, 0], [0, 0, 1; 1, 1, 0; 0, 1, 0], [0, 1, 0; 0, 0, 1; 1, 0, 1] or [0, 0, 1; 1, 0, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014

Counts closed walks of length (n+3) on a unidirectional triangle, containing a loop at one of remaining vertices. - David Neil McGrath, Sep 15 2014

a(n+2) equals the number of binary words of length n, having at least two zeros between every two successive ones. - Milan Janjic, Feb 07 2015

a(n+1)/a(n) tends to x = 1.465571... (decimal expansion given in A092526) in the limit n -> infinity. This is the real solution of x^3 - x^2 -1 = 0. See also the formula by Benoit Cloitre, Nov 30 2002. - Wolfdieter Lang, Apr 24 2015

a(n+2) equals the number of subsets of {1,2,..,n} in which any two elements differ by at least 3. - Robert FERREOL, Feb 17 2016

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x. If a positive integer N such that r = N^(1/3) is not an integer, then the number of (not necessarily distinct) integers in g(n) is A000930(n), for n >= 1. (See A274142.) - Clark Kimberling, Jun 13 2016

a(n-3) is the number of compositions of n excluding 1 and 2, n >= 3. - Gregory L. Simay, Jul 12 2016

Antidiagonal sums of array A277627. - Paul Curtz, May 16 2019

a(n+1) is the number of multus bitstrings of length n with no runs of 3 ones. - Steven Finch, Mar 25 2020

Suppose we have a(n) samples, exactly one of which is positive. Assume the cost for testing a mix of k samples is 3 if one of the samples is positive (but you will not know which sample was positive if you test more than 1) and 1 if none of the samples is positive. Then the cheapest strategy for finding the positive sample is to have a(n-3) undergo the first test and then continue with testing either a(n-4) if none were positive or with a(n-6) otherwise. The total cost of the tests will be n. - Ruediger Jehn, Dec 24 2020

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, id. 8,80.

Crilly, Tony. "A supergolden rectangle." The Mathematical Gazette 78, No. 483 (1994): 320-325. See page 234.

R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [See p. 12, line 3]

H. Langman, Play Mathematics. Hafner, NY, 1962, p. 13.

David Sankoff and Lani Haque, Power Boosts for Cluster Tests, in Comparative Genomics, Lecture Notes in Computer Science, Volume 3678/2005, Springer-Verlag. - N. J. A. Sloane, Jul 09 2009

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

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Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms, 3rd Computer Music Conference, 1996.

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

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Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).

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Russ Chamberlain, Sam Ginsburg and Chi Zhang, Generating Functions and Wilf-equivalence on Theta_k-embeddings, University of Wisconsin, April 2012.

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Eunice Y. S. Chan and Robert Corless, A new kind of companion matrix, Electronic Journal of Linear Algebra, Volume 32, Article 25, 2017, see p. 339.

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

Tony Crilly, Narayana's integer sequence revisited, Math. Gaz. (2024) Vol. 108, Issue 572, 262-269.

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

James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.

Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 11.

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - N. J. A. Sloane, Jun 10 2012

M. Feinberg, New slants, Fib. Quart. 2 (1964), 223-227.

Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373.

Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arXiv:1911.12464 [cs.FL], 2019.

I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.

Philip Gibbs and Judson McCranie, The Ulam Numbers up to One Trillion, (2017).

Maria M. Gillespie, Kenneth G. Monks, and Kenneth M. Monks, Enumerating Anchored Permutations with Bounded Gaps, arXiv:1808.03573 [math.CO], 2018.

Taras Goy, On identities with multinomial coefficients for Fibonacci-Narayana sequence, Annales Mathematicae et Informaticae (2018) 49, Eszterházy Károly University Institute of Mathematics and Informatics, 75-84.

T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 (1968), 13-21.

Russelle Guadalupe, On the 3-adic valuation of the Narayana numbers, arXiv:2112.06187 [math.NT], 2021.

R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]

R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,1).

W. R. Heinson, Simulation studies on shape and growth kinetics for fractal aggregates in aerosol and colloidal systems, PhD Dissertation, Kansas State Univ., Manhattan, Kansas, 2015; see page 49.

J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 14

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 376

Milan Janjic, Recurrence Relations and Determinants, arXiv preprint arXiv:1112.2466 [math.CO], 2011.

Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5.

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

Tom Johnson, Illustrated Music #8, Narayana’s Cows

B. Keszegh, N Lemons and D. Palvolgyi, Online and quasi-online colorings of wedges and intervals, arXiv preprint arXiv:1207.4415 [math.CO], 2012-2015.

K. Kirthi, Narayana Sequences for Cryptographic Applications, arXiv preprint arXiv:1509.05745 [math.NT], 2015.

Martin Küttler, Maksym Planeta, Jan Bierbaum, Carsten Weinhold, Hermann Härtig, Amnon Barak, and Torsten Hoefler, Corrected trees for reliable group communication, Proceedings of the 24th Symposium on Principles and Practice of Parallel Programming (PPoPP 2019), 287-299.

Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.

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Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

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

Parmanand Singh, The Ganita Kaumudi of Narayana Pandita, Ganita-Bharati, Vol. 20, No. 1-4 (1998), pp. 25-82. See pp. 79-80.

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

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E. Wilson, The Scales of Mt. Meru

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

FORMULA

G.f.: 1/(1-x-x^3). - Simon Plouffe in his 1992 dissertation

a(n) = Sum_{i=0..floor(n/3)} binomial(n-2*i, i).

a(n) = a(n-2) + a(n-3) + a(n-4) for n>3.

a(n) = floor(d*c^n + 1/2) where c is the real root of x^3-x^2-1 and d is the real root of 31*x^3-31*x^2+9*x-1 (c = 1.465571... = A092526 and d = 0.611491991950812...). - Benoit Cloitre, Nov 30 2002

a(n) = Sum_{k=0..n} binomial(floor((n+2k-2)/3), k). - Paul Barry, Jul 06 2004

a(n) = Sum_{k=0..n} binomial(k, floor((n-k)/2))(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006

a(n) = Sum_{k=0..n} binomial((n+2k)/3,(n-k)/3)*(2*cos(2*Pi*(n-k)/3)+1)/3. - Paul Barry, Dec 15 2006

a(n) = term (1,1) in matrix [1,1,0; 0,0,1; 1,0,0]^n. - Alois P. Heinz, Jun 20 2008

G.f.: exp( Sum_{n>=1} ((1+sqrt(1+4*x))^n + (1-sqrt(1+4*x))^n)*(x/2)^n/n ).

Logarithmic derivative equals A001609. - Paul D. Hanna, Oct 08 2009

a(n) = a(n-1) + a(n-2) - a(n-5) for n>4. - Paul Weisenhorn, Oct 28 2011

For n >= 2, a(2*n-1) = a(2*n-2)+a(2*n-4); a(2*n) = a(2*n-1)+a(2*n-3). - Vladimir Shevelev, Apr 12 2012

INVERT transform of (1,0,0,1,0,0,1,0,0,1,...) = (1, 1, 1, 2, 3, 4, 6, ...); but INVERT transform of (1,0,1,0,0,0,...) = (1, 1, 2, 3, 4, 6, ...). - Gary W. Adamson, Jul 05 2012

G.f.: 1/(G(0)-x) where G(k) = 1 - x^2/(1 - x^2/(x^2 - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2012

G.f.: 1 + x/(G(0)-x) where G(k) = 1 - x^2*(2*k^2 + 3*k +2) + x^2*(k+1)^2*(1 - x^2*(k^2 + 3*k +2))/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 27 2012

a(2*n) = A002478(n), a(2*n+1) = A141015(n+1), a(3*n) = A052544(n), a(3*n+1) = A124820(n), a(3*n+2) = A052529(n+1). - Johannes W. Meijer, Jul 21 2013, corrected by Greg Dresden, Jul 06 2020

G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013

a(n) = v1*w1^n+v3*w2^n+v2*w3^n, where v1,2,3 are the roots of (-1+9*x-31*x^2+31*x^3): [v1=0.6114919920, v2=0.1942540040 - 0.1225496913*I, v3=conjugate(v2)] and w1,2,3 are the roots of (-1-x^2+x^3): [w1=1.4655712319, w2=-0.2327856159 - 0.7925519925*I, w3=conjugate(w2)]. - Gerry Martens, Jun 27 2015

a(n+6)^2 + a(n+1)^2 + a(n)^2 = a(n+5)^2 + a(n+4)^2 + 3*a(n+3)^2 + a(n+2)^2. - Greg Dresden, Jul 07 2021

EXAMPLE

The number of compositions of 11 without any 1's and 2's is a(11-3) = a(8) = 13. The compositions are (11), (8,3), (3,8), (7,4), (4,7), (6,5), (5,6), (5,3,3), (3,5,3), (3,3,5), (4,4,3), (4,3,4), (3,4,4). - Gregory L. Simay, Jul 12 2016

The compositions from the above example may be mapped to the a(8) compositions of 8 into 1's and 3's using this (more generally applicable) method: replace all numbers greater than 3 with a 3 followed by 1's to make the same total, then remove the initial 3 from the composition. Maintaining the example's order, they become (1,1,1,1,1,1,1,1), (1,1,1,1,1,3), (3,1,1,1,1,1), (1,1,1,1,3,1), (1,3,1,1,1,1), (1,1,1,3,1,1), (1,1,3,1,1,1), (1,1,3,3), (3,1,1,3), (3,3,1,1), (1,3,1,3), (1,3,3,1), (3,1,3,1). - Peter Munn, May 31 2017

MAPLE

f := proc(r) local t1, i; t1 := []; for i from 1 to r do t1 := [op(t1), 0]; od: for i from 1 to r+1 do t1 := [op(t1), 1]; od: for i from 2*r+2 to 50 do t1 := [op(t1), t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 2)}, unlabeled]: seq(count(SeqSetU, size=j), j=3..40); # Zerinvary Lajos, Oct 10 2006

A000930 := proc(n)

add(binomial(n-2*k, k), k=0..floor(n/3)) ;

end proc: # Zerinvary Lajos, Apr 03 2007

a:= n-> (Matrix([[1, 1, 0], [0, 0, 1], [1, 0, 0]])^n)[1, 1]: seq(a(n), n=0..50); # Alois P. Heinz, Jun 20 2008

MATHEMATICA

a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 3]; Table[ a[n], {n, 0, 40} ]

CoefficientList[Series[1/(1-x-x^3), {x, 0, 45}], x] (* Zerinvary Lajos, Mar 22 2007 *)

LinearRecurrence[{1, 0, 1}, {1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

a[n_] := HypergeometricPFQ[{(1-n)/3, (2-n)/3, -n/3}, {(1-n)/ 2, -n/2}, -27/4]; Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Feb 26 2013 *)

PROG

(PARI) a(n)=polcoeff(exp(sum(m=1, n, ((1+sqrt(1+4*x))^m + (1-sqrt(1+4*x))^m)*(x/2)^m/m)+x*O(x^n)), n) \\ Paul D. Hanna, Oct 08 2009

(PARI) x='x+O('x^66); Vec(1/(1-(x+x^3))) \\ Joerg Arndt, May 24 2011

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, 0, 1]^n*[1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Feb 26 2017

(Maxima) makelist(sum(binomial(n-2*k, k), k, 0, n/3), n, 0, 18); \\ Emanuele Munarini, May 24 2011

(Haskell)

a000930 n = a000930_list !! n

a000930_list = 1 : 1 : 1 : zipWith (+) a000930_list (drop 2 a000930_list)

-- Reinhard Zumkeller, Sep 25 2011

(Magma) [1, 1] cat [ n le 3 select n else Self(n-1)+Self(n-3): n in [1..50] ]; // Vincenzo Librandi, Apr 25 2015

(GAP) a:=[1, 1, 1];; for n in [4..50] do a[n]:=a[n-1]+a[n-3]; od; a; # Muniru A Asiru, Aug 13 2018

(Python)

from itertools import islice

def A000930_gen(): # generator of terms

blist = [1]*3

while True:

yield blist[0]

blist = blist[1:]+[blist[0]+blist[2]]

(SageMath)

@CachedFunction

def a(n): # A000930

if (n<3): return 1

else: return a(n-1) + a(n-3)

[a(n) for n in (0..80)] # G. C. Greubel, Jul 29 2022

CROSSREFS

A120562 has the same recurrence for odd n.

KEYWORD

nonn,easy,nice

AUTHOR

EXTENSIONS

Name expanded by N. J. A. Sloane, Sep 07 2012

STATUS

approved