This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000930 M0571 N0207 #617 Aug 15 2025 06:01:20
%S A000930 1,1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406,595,872,1278,1873,
%T A000930 2745,4023,5896,8641,12664,18560,27201,39865,58425,85626,125491,
%U A000930 183916,269542,395033,578949,848491,1243524,1822473,2670964,3914488,5736961,8407925
%N A000930 Narayana's cows sequence: a(0) = a(1) = a(2) = 1; thereafter a(n) = a(n-1) + a(n-3).
%C A000930 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]
%C A000930 Number of compositions of n into parts 1 and 3. - _Joerg Arndt_, Jun 25 2011
%C A000930 A Lamé sequence of higher order.
%C A000930 Could have begun 1,0,0,1,1,1,2,3,4,6,9,... (A078012) but that would spoil many nice properties.
%C A000930 Number of tilings of a 3 X n rectangle with straight trominoes.
%C A000930 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.
%C A000930 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]
%C A000930 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.
%C A000930 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]
%C A000930 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
%C A000930 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
%C A000930 Row sums of Riordan array (1/(1-x^3), x/(1-x^3)). - _Paul Barry_, Feb 25 2005
%C A000930 Row sums of Riordan array (1,x(1+x^2)). - _Paul Barry_, Jan 12 2006
%C A000930 Starting with offset 1 = row sums of triangle A145580. - _Gary W. Adamson_, Oct 13 2008
%C A000930 Number of digits in A061582. - _Dmitry Kamenetsky_, Jan 17 2009
%C A000930 From _Jon Perry_, Nov 15 2010: (Start)
%C A000930 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):
%C A000930 1 1 1 1 1 1 1 1 1 1 1 1 1
%C A000930 1 2 3 4 5 6 7 8 9 10
%C A000930 1 3 6 10 15 21 28
%C A000930 1 4 10 20
%C A000930 1
%C A000930 ------------------------------
%C A000930 1 1 1 2 3 4 6 9 13 19 28 41 60
%C A000930 with (in this case 3) leading zeros added to each row.
%C A000930 (End)
%C A000930 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
%C A000930 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
%C A000930 For n>=2, row sums of Pascal's triangle (A007318) with triplicated diagonals. - _Vladimir Shevelev_, Apr 12 2012
%C A000930 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
%C A000930 Diagonal sums of triangle A011973. - _John Molokach_, Jul 06 2013
%C A000930 "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
%C A000930 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
%C A000930 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
%C A000930 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
%C A000930 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
%C A000930 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
%C A000930 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
%C A000930 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
%C A000930 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
%C A000930 a(n-3) is the number of compositions of n excluding 1 and 2, n >= 3. - _Gregory L. Simay_, Jul 12 2016
%C A000930 Antidiagonal sums of array A277627. - _Paul Curtz_, May 16 2019
%C A000930 a(n+1) is the number of multus bitstrings of length n with no runs of 3 ones. - _Steven Finch_, Mar 25 2020
%C A000930 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
%D A000930 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, id. 8,80.
%D A000930 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]
%D A000930 H. Langman, Play Mathematics. Hafner, NY, 1962, p. 13.
%D A000930 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
%D A000930 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000930 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000930 Harvey P. Dale and T. D. Noe, <a href="/A000930/b000930.txt">Table of n, a(n) for n = 0..5000</a> [The first 500 terms were computed by T. D. Noe]
%H A000930 Gene Abrams, Stefan Erickson, and Cristóbal Gil Canto, <a href="https://arxiv.org/abs/1712.06480">Leavitt path algebras of Cayley graphs C_n^j</a>, arXiv:1712.06480 [math.RA], 2017.
%H A000930 Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, <a href="https://www.emis.de/journals/JIS/VOL22/Shattuck/shattuck13.html">Some New Restricted n-Color Composition Functions</a>, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
%H A000930 Mudit Aggarwal and Samrith Ram, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Ram/ram3.html">Generating Functions for Straight Polyomino Tilings of Narrow Rectangles</a>, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
%H A000930 J.-P. Allouche and T. Johnson, <a href="https://hal.science/hal-02986050v1">Narayana's cows and delayed morphisms</a>, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
%H A000930 J.-P. Allouche and T. Johnson, <a href="/A000930/a000930.pdf">Narayana's cows and delayed morphisms</a>, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
%H A000930 I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, <a href="http://arxiv.org/abs/1509.05239">Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences</a>, arXiv:1509.05239v1 [math.CO] Sep 17 2015. See Conjecture 5.8.
%H A000930 Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H A000930 Bruce M. Boman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/58-5/boman.pdf">Geometric Capitulum Patterns Based on Fibonacci p-Proportions</a>, Fibonacci Quart. 58 (2020), no. 5, 91-102.
%H A000930 Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-5/Boman.pdf">Why do Fibonacci numbers appear in patterns of growth in nature?</a>, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
%H A000930 Eric Fernando Bravo, <a href="https://www.mathos.unios.hr/mc/index.php/mc/article/view/4733">On concatenations of Padovan and Perrin numbers</a>, Math. Commun. (2023) Vol 28, 105-119.
%H A000930 Russ Chamberlain, Sam Ginsburg and Chi Zhang, <a href="http://digital.library.wisc.edu/1793/61870">Generating Functions and Wilf-equivalence on Theta_k-embeddings</a>, University of Wisconsin, April 2012.
%H A000930 Eunice Y. S. Chan, <a href="http://ir.lib.uwo.ca/etd/4028">A comparison of solution methods for Mandelbrot-like polynomials</a>, MS Thesis, 2016, University of Western Ontario; Electronic Thesis and Dissertation Repository. 4028.
%H A000930 Eunice Y. S. Chan, <a href="https://ir.lib.uwo.ca/etd/6414">Algebraic Companions and Linearizations</a>, The University of Western Ontario (Canada, 2019) Electronic Thesis and Dissertation Repository. 6414.
%H A000930 Eunice Y. S. Chan and Robert Corless, <a href="https://doi.org/10.13001/1081-3810.3400">A new kind of companion matrix</a>, Electronic Journal of Linear Algebra, Volume 32, Article 25, 2017, see p. 339.
%H A000930 P. Chinn and S. Heubach, <a href="/A005710/a005710.pdf">(1, k)-compositions</a>, Congr. Numer. 164 (2003), 183-194. [Local copy]
%H A000930 Hùng Việt Chu, <a href="https://arxiv.org/abs/2005.10081">Various sequences from counting subsets</a>, version 2, June 2021.
%H A000930 Johann Cigler, <a href="https://arxiv.org/abs/1912.06651">Some remarks on generalized Fibonacci and Lucas polynomials</a>, arXiv:1912.06651 [math.CO], 2019.
%H A000930 Tony Crilly, <a href="https://www.jstor.org/stable/3620208">A supergolden rectangle</a>, The Mathematical Gazette 78, No. 483 (1994): 320-325. See page 234.
%H A000930 Tony Crilly, <a href="https://doi.org/10.1017/mag.2024.65">Narayana's integer sequence revisited</a>, Math. Gaz. (2024) Vol. 108, Issue 572, 262-269.
%H A000930 E. Di Cera and Y. Kong, <a href="http://dx.doi.org/10.1016/S0301-4622(96)02178-3">Theory of multivalent binding in one and two-dimensional lattices</a>, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%H A000930 James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H A000930 Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, <a href="https://arxiv.org/abs/2310.06288">Catalan-Spitzer permutations</a>, arXiv:2310.06288 [math.CO], 2023. See p. 11.
%H A000930 Larry Ericksen and Peter G. Anderson, <a href="http://www.cs.rit.edu/~pga/k-zeck.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, February 2012. - _N. J. A. Sloane_, Jun 10 2012
%H A000930 M. Feinberg, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-3/feinberg.pdf">New slants</a>, Fib. Quart. 2 (1964), 223-227.
%H A000930 Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H A000930 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 373.
%H A000930 Lukas Fleischer and Jeffrey Shallit, <a href="https://arxiv.org/abs/1911.12464">Words With Few Palindromes, Revisited</a>, arXiv:1911.12464 [cs.FL], 2019.
%H A000930 I. M. Gessel and Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5.
%H A000930 Philip Gibbs and Judson McCranie, <a href="https://www.researchgate.net/profile/Philip_Gibbs/publication/320980165_The_Ulam_Numbers_up_to_One_Trillion/links/5a058786aca2726b4c78588d/The-Ulam-Numbers-up-to-One-Trillion.pdf">The Ulam Numbers up to One Trillion</a>, (2017).
%H A000930 Maria M. Gillespie, Kenneth G. Monks, and Kenneth M. Monks, <a href="https://arxiv.org/abs/1808.03573">Enumerating Anchored Permutations with Bounded Gaps</a>, arXiv:1808.03573 [math.CO], 2018.
%H A000930 Taras Goy, <a href="https://doi.org/10.33039/ami.2018.09.001">On identities with multinomial coefficients for Fibonacci-Narayana sequence</a>, Annales Mathematicae et Informaticae (2018) 49, Eszterházy Károly University Institute of Mathematics and Informatics, 75-84.
%H A000930 T. M. Green, <a href="http://www.jstor.org/stable/2687953">Recurrent sequences and Pascal's triangle</a>, Math. Mag., 41 (1968), 13-21.
%H A000930 Russelle Guadalupe, <a href="https://arxiv.org/abs/2112.06187">On the 3-adic valuation of the Narayana numbers</a>, arXiv:2112.06187 [math.NT], 2021.
%H A000930 R. K. Guy, <a href="/A005251/a005251_1.pdf">Anyone for Twopins?</a>, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
%H A000930 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>, Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H A000930 V. C. Harris and C. C. Styles, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,1).
%H A000930 W. R. Heinson, <a href="http://hdl.handle.net/2097/19714">Simulation studies on shape and growth kinetics for fractal aggregates in aerosol and colloidal systems</a>, PhD Dissertation, Kansas State Univ., Manhattan, Kansas, 2015; see page 49.
%H A000930 J. Hermes, <a href="http://dx.doi.org/10.1007/BF01446684">Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden</a>, Math. Ann., 45 (1894), 371-380.
%H A000930 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=14">Encyclopedia of Combinatorial Structures 14</a>
%H A000930 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=376">Encyclopedia of Combinatorial Structures 376</a>
%H A000930 Milan Janjic, <a href="http://arxiv.org/abs/1112.2466">Recurrence Relations and Determinants</a>, arXiv preprint arXiv:1112.2466 [math.CO], 2011.
%H A000930 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Janjic/janjic42.html">Determinants and Recurrence Sequences</a>, Journal of Integer Sequences, 2012, Article 12.3.5.
%H A000930 Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 91.
%H A000930 Tom Johnson, <a href="https://www.youtube.com/watch?v=VOS3piSMS9E">Illustrated Music #8, Narayana's Cows</a>
%H A000930 B. Keszegh, N Lemons and D. Palvolgyi, <a href="http://arxiv.org/abs/1207.4415">Online and quasi-online colorings of wedges and intervals</a>, arXiv preprint arXiv:1207.4415 [math.CO], 2012-2015.
%H A000930 K. Kirthi, <a href="http://arxiv.org/abs/1509.05745">Narayana Sequences for Cryptographic Applications</a>, arXiv preprint arXiv:1509.05745 [math.NT], 2015.
%H A000930 Martin Küttler, Maksym Planeta, Jan Bierbaum, Carsten Weinhold, Hermann Härtig, Amnon Barak, and Torsten Hoefler, <a href="https://doi.org/10.1145/3293883.3295721">Corrected trees for reliable group communication</a>, Proceedings of the 24th Symposium on Principles and Practice of Parallel Programming (PPoPP 2019), 287-299.
%H A000930 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Roby/roby4.html">Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
%H A000930 K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, <a href="http://arxiv.org/abs/1510.01952">Equivalence classes of ballot paths modulo strings of length 2 and 3</a>, arXiv:1510.01952 [math.CO], 2015.
%H A000930 T. Mansour and M. Shattuck, <a href="http://dx.doi.org/10.1016/j.amc.2012.12.052">Polynomials whose coefficients are generalized Tribonacci numbers</a>, Applied Mathematics and Computation, Volume 219, Issue 15, Apr 01 2013, Pages 8366-8374.
%H A000930 T. Mansour and M. Shattuck, <a href="http://arxiv.org/abs/1410.6943">A monotonicity property for generalized Fibonacci sequences</a>, arXiv preprint arXiv:1410.6943 [math.CO], 2014.
%H A000930 R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings</a>, arXiv:1311.6135 [math.CO], 2013, Table 17.
%H A000930 R. J. Mathar, <a href="http://arxiv.org/abs/1609.03964">Tiling n x m rectangles with 1 x 1 and s x s squares</a>, arXiv:1609.03964 [math.CO] (2016), Section 4.2.
%H A000930 Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
%H A000930 Denis Neiter and Amsha Proag, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Proag/proag3.html">Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.8.3.
%H A000930 J. Noonan and D. Zeilberger, <a href="http://arxiv.org/abs/math/9806036">The Goulden-Jackson Cluster Method: Extensions, Applications and Implementations</a>, arXiv:math/9806036 [math.CO], Jun 08 1998.
%H A000930 Antonio M. Oller-Marcén, <a href="http://www.emis.de/journals/INTEGERS/papers/j11/j11.Abstract.html">The Dying Rabbit Problem Revisited</a>, INTEGERS, 9 (2009), 129-138.
%H A000930 Pagdame Tiebekabe and Kouèssi Norbert Adédji, <a href="https://arxiv.org/abs/2304.00773">Narayana's cows numbers which are concatenations of three repdigits in base Rho</a>, arXiv:2304.00773 [math.NT], 2023.
%H A000930 Dömötör Pálvölgyi, <a href="https://domotorp.web.elte.hu/cikkek/nagydoktori_jav.pdf">Coloring Geometric Hypergraphs</a>, Ph. D. Dissertation, Hungarian Acad. Sci. (Hungary, 2023). See p. 35.
%H A000930 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000930 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A000930 Abhishek Raj, Vadim Oganesyan, and Antonello Scardicchio, <a href="https://arxiv.org/abs/2412.05231">A kinetically constrained model exhibiting non-linear diffusion and jamming</a>, arXiv:2412.05231 [cond-mat.stat-mech], 2024. See p. 8.
%H A000930 Narad Rampersad and Max Wiebe, <a href="https://math.colgate.edu/~integers/y73/y73.pdf">Sums of products of binomial coefficients mod 2 and 2-regular sequences</a>, Integers (2024) Vol. 24, Art. No. A73. See p. 9.
%H A000930 M. Randic, D. Morales, and O. Araujo, <a href="http://www.emis.de/journals/DM/v16-2/art3.pdf">Higher-order Lucas numbers</a>, Divulgaciones Matematicas 6:2 (2008), pp. 275-283.
%H A000930 Frank Ruskey and Jennifer Woodcock, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r126">Counting Fixed-Height Tatami Tilings</a>, Electronic Journal of Combinatorics, Paper R126 (2009), 20 pages.
%H A000930 Jeffrey Shallit, <a href="https://arxiv.org/abs/2503.01026">The Narayana Morphism and Related Words</a>, arXiv:2503.01026 [math.CO], 2025.
%H A000930 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series001">The binary form of Conway's Sequence</a>
%H A000930 Parmanand Singh, <a href="https://archive.org/details/ganitakaumudinarayanapanditach113ed.parmanandsingh_680_X/page/n53/mode/2up">The Ganita Kaumudi of Narayana Pandita</a>, Ganita-Bharati, Vol. 20, No. 1-4 (1998), pp. 25-82. See pp. 79-80.
%H A000930 Z. Skupien, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.003">Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles</a>, Discr. Math., 309 (2009), 6382-6390. - _N. J. A. Sloane_, Feb 12 2010
%H A000930 Yüksel Soykan, <a href="https://doi.org/10.34198/ejms.15425.605638">On Generalized co-Narayana Numbers</a>, Earthline Journal of Mathematical Sciences, 15(4), 605-638, (2025). See p. 607.
%H A000930 Krzysztof Strasburger, <a href="http://dx.doi.org/10.1063/1.4953677">The order of three lowest-energy states of the six-electron harmonium at small force constant</a>, The Journal of Chemical Physics 144, 234304 (2016).
%H A000930 Krzysztof Strasburger, <a href="https://arxiv.org/abs/1903.06051">Explicitly correlated wavefunctions of the ground state and the lowest 5-tuple state of the carbon atom</a>, arXiv:1903.06051 [physics.chem-ph], 2019.
%H A000930 Krzysztof Strasburger, <a href="https://arxiv.org/abs/2009.08723">Energy difference between the lowest doublet and quartet states of the boron atom</a>, arXiv:2009.08723 [physics.chem-ph], 2020.
%H A000930 Liam Taylor-West, <a href="https://doi.org/10.24379/RCM.00002347">Modularity, technology, and geometry in compositional practice: a portfolio of original compositions</a>, D. mus. dissertation, Royal College of Music (London, England 2023), p. 56.
%H A000930 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NarayanaCowSequence.html">Narayana Cow Sequence</a>.
%H A000930 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales of Mt. Meru</a>
%H A000930 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1).
%F A000930 G.f.: 1/(1-x-x^3). - _Simon Plouffe_ in his 1992 dissertation
%F A000930 a(n) = Sum_{i=0..floor(n/3)} binomial(n-2*i, i).
%F A000930 a(n) = a(n-2) + a(n-3) + a(n-4) for n>3.
%F A000930 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
%F A000930 a(n) = Sum_{k=0..n} binomial(floor((n+2k-2)/3), k). - _Paul Barry_, Jul 06 2004
%F A000930 a(n) = Sum_{k=0..n} binomial(k, floor((n-k)/2))(1+(-1)^(n-k))/2. - _Paul Barry_, Jan 12 2006
%F A000930 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
%F A000930 a(n) = term (1,1) in matrix [1,1,0; 0,0,1; 1,0,0]^n. - _Alois P. Heinz_, Jun 20 2008
%F A000930 G.f.: exp( Sum_{n>=1} ((1+sqrt(1+4*x))^n + (1-sqrt(1+4*x))^n)*(x/2)^n/n ).
%F A000930 Logarithmic derivative equals A001609. - _Paul D. Hanna_, Oct 08 2009
%F A000930 a(n) = a(n-1) + a(n-2) - a(n-5) for n>4. - _Paul Weisenhorn_, Oct 28 2011
%F A000930 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
%F A000930 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
%F A000930 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
%F A000930 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
%F A000930 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
%F A000930 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
%F A000930 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
%F A000930 a(n) = (6*A001609(n+3) + A001609(n-7))/31 for n>=7. - _Areebah Mahdia_, Jun 07 2020
%F A000930 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
%F A000930 a(n) = Sum_{i=(n-7)..(n-1)} a(i) / 2. - _Jules Beauchamp_, May 10 2025
%e A000930 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
%e A000930 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
%p A000930 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
%p A000930 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
%p A000930 A000930 := proc(n)
%p A000930 add(binomial(n-2*k,k),k=0..floor(n/3)) ;
%p A000930 end proc: # _Zerinvary Lajos_, Apr 03 2007
%p A000930 a:= n-> (<<1|1|0>, <0|0|1>, <1|0|0>>^n)[1,1]:
%p A000930 seq(a(n), n=0..50); # _Alois P. Heinz_, Jun 20 2008
%t A000930 a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 3]; Table[ a[n], {n, 0, 40} ]
%t A000930 CoefficientList[Series[1/(1 - x - x^3), {x, 0, 45}], x] (* _Zerinvary Lajos_, Mar 22 2007 *)
%t A000930 LinearRecurrence[{1, 0, 1}, {1, 1, 1}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 11 2012 *)
%t A000930 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 *)
%t A000930 Table[-RootSum[1 + #^2 - #^3 &, 3 #^(n + 2) - 11 #^(n + 3) + 2 #^(n + 4) &]/31, {n, 20}] (* _Eric W. Weisstein_, Feb 14 2025 *)
%o A000930 (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
%o A000930 (PARI) x='x+O('x^66); Vec(1/(1-(x+x^3))) \\ _Joerg Arndt_, May 24 2011
%o A000930 (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
%o A000930 (Maxima) makelist(sum(binomial(n-2*k,k),k,0,n/3),n,0,18); /* _Emanuele Munarini_, May 24 2011 */
%o A000930 (Haskell)
%o A000930 a000930 n = a000930_list !! n
%o A000930 a000930_list = 1 : 1 : 1 : zipWith (+) a000930_list (drop 2 a000930_list)
%o A000930 -- _Reinhard Zumkeller_, Sep 25 2011
%o A000930 (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
%o A000930 (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
%o A000930 (Python)
%o A000930 from itertools import islice
%o A000930 def A000930_gen(): # generator of terms
%o A000930 blist = [1]*3
%o A000930 while True:
%o A000930 yield blist[0]
%o A000930 blist = blist[1:]+[blist[0]+blist[2]]
%o A000930 A000930_list = list(islice(A000930_gen(),30)) # _Chai Wah Wu_, Feb 04 2022
%o A000930 (SageMath)
%o A000930 @CachedFunction
%o A000930 def a(n): # A000930
%o A000930 if (n<3): return 1
%o A000930 else: return a(n-1) + a(n-3)
%o A000930 [a(n) for n in (0..80)] # _G. C. Greubel_, Jul 29 2022
%Y A000930 For Lamé sequences of orders 1 through 9 see A000045, this sequence, and A017898 - A017904.
%Y A000930 Cf. A000073, A000213, A048715, A069241, A092526, A170954.
%Y A000930 See also A000079, A003269, A003520, A005708, A005709, A005710.
%Y A000930 Essentially the same as A068921 and A078012.
%Y A000930 See also A001609, A145580, A179070, A214551 (same rule except divide by GCD).
%Y A000930 A271901 and A271953 give the period of this sequence mod n.
%Y A000930 Cf. A007318, A060576, A102547, A277627, A289207.
%Y A000930 A120562 has the same recurrence for odd n.
%K A000930 nonn,easy,nice
%O A000930 0,4
%A A000930 _N. J. A. Sloane_
%E A000930 Name expanded by _N. J. A. Sloane_, Sep 07 2012