A131577 -id:A131577 - cp's OEIS frontend

A005418 Number of (n-1)-bead black-white reversible strings; also binary grids; also row sums of Losanitsch's triangle A034851; also number of caterpillar graphs on n+2 vertices.

1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, 8256, 16512, 32896, 65792, 131328, 262656, 524800, 1049600, 2098176, 4196352, 8390656, 16781312, 33558528, 67117056, 134225920, 268451840, 536887296, 1073774592, 2147516416, 4295032832

Offset: 1

N. J. A. Sloane, R. K. Guy

Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, n "corners"; the symmetry group is S3, reversing a walk does not count as different. Walks are not self-avoiding. - Colin Mallows
Slavik V. Jablan observes that this is also the number of rational knots and links with n+2 crossings (cf. A018240). See reference. [Corrected by Andrey Zabolotskiy, Jun 18 2020]
Number of bit strings of length (n-1), not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different. - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001
The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi, May 14 2007
Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal, see example. - Franklin T. Adams-Watters, Oct 24 2009
Number of normally non-isomorphic realizations of the associahedron of type I starting with dimension 2 in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of fibonacenes with n+2 hexagons. See the Balaban and the Dobrynin references. - Emeric Deutsch, Apr 21 2013
From the point of view of binary grids, it is a (1,n)-rectangular grid. A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11. - Yosu Yurramendi, May 19 2013
Number of n-vertex difference graphs (bipartite 2K_2-free graphs) [Peled & Sun, Thm. 9]. - Falk Hüffner, Jan 10 2016
The offset should be 0, since the first row of A034851 is row 0. The name would then be: "Number of n bead...". - Daniel Forgues, Jul 26 2018
a(n) is the number of non-isomorphic generalized rigid ladders with n cells. A generalized rigid ladder with n cells is a graph with vertex set is the union of {u_0, u_1, ..., u_n} and {v_0, v_1, ..., v_n}, and for every 0 <= i <= n-1, the edges are of the form {u_i,u_i+1}, {v_i, v_i+1}, {u_i,v_i} and either {u_i,v_i+1} or {u_i+1,v_i}. - Christian Barrientos, Jul 29 2018
Also number of non-isomorphic stairs with n+1 cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. - Christian Barrientos and Sarah Minion, Jul 29 2018
From Robert A. Russell, Oct 28 2018: (Start)
There are two different unoriented row colorings using two colors that give us very similar results here, a difference of one in the offset. In an unoriented row, chiral pairs are counted as one.
a(n) is the number of color patterns (set partitions) of an unoriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if the colors are permutable.
a(n+1) is the number of ways to color an unoriented row of length n using two noninterchangeable colors (one need not use both colors).
See the examples below of these two different colorings. (End)
Also arises from the enumeration of types of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020
a(n) is the number of (unlabeled) 2-paths with n+4 vertices. (A 2-path with order n at least 4 can be constructed from a 3-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to an existing 2-clique containing an existing 2-leaf.) - Allan Bickle, Apr 05 2022
a(n) is the number of caterpillars with a perfect matching and order 2n+2. - Christian Barrientos, Sep 12 2023
a(n) is also the number of distinct planar embeddings of the (n+2)-centipede graph (up to at least n=8 and likely for all larger n). - Eric W. Weisstein, May 21 2024
a(n) is also the number of distinct planar embeddings of the 2 X (n+2) grid graph i.e., the (n+2)-ladder graph. - Eric W. Weisstein, May 21 2024
Dimension of the homogeneous component of degree n of the free Jordan algebra on two generators (or, in this case, the free special Jordan algebra on two generators). It follows from (Shirshov 1956, Cohn 1959). - Vladimir Dotsenko, Mar 29 2025

			a(5) = 10 because there are 16 compositions of 5 (shown as <vectors>) but only 10 equivalence classes (shown as {sets}): {<5>}, {<4,1>,<1,4>}, {<3,2>,<2,3>}, {<3,1,1>,<1,1,3>}, {<1,3,1>},{<2,2,1>,<1,2,2>}, {<2,1,2>}, {<2,1,1,1>,<1,1,1,2>}, {<1,2,1,1>,<1,1,2,1>}, {<1,1,1,1,1>}. - _Geoffrey Critzer_, Nov 02 2012
G.f. = x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 36*x^7 + 72*x^8 + ... - _Michael Somos_, Jun 24 2018
From _Robert A. Russell_, Oct 28 2018: (Start)
For a(5)=10, the 4 achiral patterns (set partitions) are AAAAA, AABAA, ABABA, and ABBBA. The 6 chiral pairs are AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB. The colors are permutable.
For n=4 and a(n+1)=10, the 4 achiral colorings are AAAA, ABBA, BAAB, and BBBB. The 6 achiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB. The colors are not permutable. (End)

K. Balasubramanian, "Combinatorial Enumeration of Chemical Isomers", Indian J. Chem., (1978) vol. 16B, pp. 1094-1096. See page 1095.
Wayne M. Dymacek, Steinhaus graphs. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 399--412, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561065 (81f:05120)
Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
Joseph S. Madachy: Madachy's Mathematical Recreations. New York: Dover Publications, Inc., 1979, p. 46 (first publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation)
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
C. A. Pickover, Keys to Infinity, Wiley 1995, p. 75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 151 and 733.
Andrei Asinowski and Alon Regev, Triangulations with Few Ears: Symmetry Classes and Disjointness, Integers 16 (2016), #A5.
A. T. Balaban, Chemical graphs. Part 50. Symmetry and enumeration of fibonacenes (unbranched catacondensed benzenoids isoarithmic with helicenes and zigzag catafusenes), MATCH: Commun. Math. Comput. Chem., 24 (1989) 29-38.
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. P. Burger, M. Van Der Merwe, and J. H. Van Vuuren, An asymptotic analysis of the evolutionary spatial prisoner's dilemma on a path, Discrete Appl. Math. 160, No. 15, 2075-2088 (2012), Table 3.1.
C. Ceballos, F. Santos, and G. Ziegler, Many Non-equivalent Realizations of the Associahedron, arXiv:1109.5544 [math.MG], 2011-2013; pp. 15 and 26.
P. M. Cohn, Two embedding theorems for Jordan algebras, Proceedings of the London Mathematical Society, Volume s3-9, Issue 4, October 1959, pp. 503-524.
Jacob Crabtree, Another Enumeration of Caterpillar Trees, arXiv:1810.11744 [math.CO], 2018.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, E. Brendsdal, Zhang Fuji, Guo Xiaofeng, and R. Tosic, Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
Miroslav Marinov Dimitrov, Designing Boolean Functions and Digital Sequences for Cryptology and Communications, Ph. D. Dissertation, Bulgarian Acad. Sci. (Sofia, Bulgaria 2023).
A. A. Dobrynin, On the Wiener index of fibonacenes, MATCH: Commun. Math. Comput. Chem, 64 (2010), 707-726.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
Sahir Gill, Bounds for Region Containing All Zeros of a Complex Polynomial, International Journal of Mathematical Analysis (2018), Vol. 12, No. 7, 325-333.
T. A. Gittings, Minimum braids: a complete invariant of knots and links, arXiv:math/0401051 [math.GT], 2004. - _N. J. A. Sloane_, Jan 18 2013
R. K. Guy, Letter to N. J. A. Sloane, Nov 1978.
Frank Harary and Allen J. Schwenk, The number of caterpillars, Discrete Mathematics, Volume 6, Issue 4, 1973, 359-365.
N. Hoffman, Binary grids and a related counting problem, Two-Year College Math. J. 9 (1978), 267-272.
S. V. Jablan, Geometry of Links, XII Yugoslav Geometric Seminar (Novi Sad, 1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Isaac B. Michael and M. R. Sepanski, Net regular signed trees, Australasian Journal of Combinatorics, Volume 66(2) (2016), 192-204.
U. N. Peled and F. Sun, Enumeration of difference graphs, Discrete Appl. Math., 60 (1995), 311-318.
Paulo Renato da Costa Pereira, Lilian Markenzon and Oswaldo Vernet, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
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
Hans Rademacher, On the number of certain types of polyhedra, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564. See Theorem 8, Eq. 14.3.
A. Regev, Remarks on two-eared triangulations, arXiv preprint arXiv:1309.0743 [math.CO], 2013-2014.
Suthee Ruangwises, The Landscape of Computing Symmetric n-Variable Functions with 2n Cards, arXiv:2306.13551 [cs.CR], 2023.
A. I. Shirshov, On special J-rings, Mat. Sb. (N.S.), 38(80), 1956, pp. 149-166.
N. J. A. Sloane, Classic Sequences
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), Article #00.1.1.
Eric Weisstein's World of Mathematics, Barker Code.
Eric Weisstein's World of Mathematics, Bishops Problem.
Eric Weisstein's World of Mathematics, Caterpillar Graph.
Eric Weisstein's World of Mathematics, Centipede Graph.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Ladder Graph.
Eric Weisstein's World of Mathematics, Losanitsch's Triangle.
Eric Weisstein's World of Mathematics, Planar Embedding.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 87 (2014), 1260-1264; see Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Column 2 of A320750 (set partitions).
Cf. A131577 (oriented), A122746(n-3) (chiral), A016116 (achiral), for set partitions with up to two subsets.
Column 2 of A277504, offset by one (colors not permutable).
Cf. A000079 (oriented), A122746(n-2) (chiral), and A060546 (achiral), for a(n+1).
Cf. A001998, A001444, A051436, A051437, A007582, A001445, A032085.

a005418 n = sum $ a034851_row (n - 1) -- Reinhard Zumkeller, Jan 14 2012

A005418 := n->2^(n-2)+2^(floor(n/2)-1): seq(A005418(n), n=1..34);

LinearRecurrence[{2,2,-4}, {1,2,3}, 40] (* or *) Table[2^(n-2)+2^(Floor[n/2]-1), {n,40}] (* Harvey P. Dale, Jan 18 2012 *)

A005418(n)= 2^(n-2) + 2^(n\2-1); \\ Joerg Arndt, Sep 16 2013

def A005418(n): return 1 if n == 1 else 2**((m:= n//2)-1)*(2**(n-m-1)+1) # Chai Wah Wu, Feb 03 2022

a(n) = 2^(n-2) + 2^(floor(n/2) - 1).
G.f.: -x*(-1 + 3*x^2) / ( (2*x - 1)*(2*x^2 - 1) ). - Simon Plouffe in his 1992 dissertation
G.f.: x*(1+2*x)*(1-3*x^2)/((1-4*x^2)*(1-2*x^2)), not reduced. - Wolfdieter Lang, May 08 2001
a(n) = 6*a(n - 2) - 8*a(n - 4). a(2*n) = A063376(n - 1) = 2*a(2*n - 1); a(2*n + 1) = A007582(n). - Henry Bottomley, Jul 14 2001
a(n+2) = 2*a(n+1) - A077957(n) with a(1) = 1, a(2) = 2. - Yosu Yurramendi, Oct 24 2008
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Jaume Oliver Lafont, Dec 05 2008
Union of A007582 and A161168. Union of A007582 and A063376. - Jaroslav Krizek, Aug 14 2009
G.f.: G(0); G(k) = 1 + 2*x/(1 - x*(1+2^(k+1))/(x*(1+2^(k+1)) + (1+2^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 12 2011
a(2*n) = 2*a(2*n-1) and a(2*n+1) = a(2*n) + 4^(n-1) with a(1) = 1. - Johannes W. Meijer, Aug 26 2013
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A131577(n) + A016116(n)) / 2 = A131577(n) - A122746(n-3) = A122746(n-3) + A016116(n), for set partitions with up to two subsets.
a(n+1) = (A000079(n) + A060546(n)) / 2 = A000079(n) - A122746(n-2) = A122746(n-2) + A060546(n), for two colors that do not permute.
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=2 is the maximum number of colors, S2(n,k) is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n+1) = (k^n + k^ceiling(n/2)) / 2, where k=2 is number of colors we can use. (End)
E.g.f.: (cosh(2*x) + 2*cosh(sqrt(2)*x) + sinh(2*x) + sqrt(2)*sinh(sqrt(2)*x) - 3)/4. - Stefano Spezia, Jun 01 2022

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, 683, 1365, 2730, 5461, 10923, 21846, 43691, 87381, 174762, 349525, 699051, 1398102, 2796203, 5592405, 11184810, 22369621, 44739243, 89478486, 178956971, 357913941, 715827882, 1431655765, 2863311531, 5726623062

Offset: 0

Clark Kimberling

Trisections give A082365, A132804, A132805. - Paul Curtz, Nov 18 2007
If the offset is changed to 1, this is the maximal number of closed regions bounded by straight lines after n straight line cuts in a plane: a(n) = a(n-1) + n - 3, a(1)=0; a(2)=0; a(3)=1; and so on. - Srikanth K S, Jan 23 2008
M^n * [1,0,0] = [A024493(n), a(n), A024494(n)]; where M = a 3x3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - Gary W. Adamson, Mar 13 2009
For n>=1, a(n-1) is the number of generalized compositions of n when there are i^2/2 - 3*i/2 + 1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n) + A024494(n)*M + a(n)*M^2. - Stanislav Sykora, Jun 10 2012
{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017
This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^3; see A291000.  - Clark Kimberling, Aug 24 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2, example 9.
Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(2) = a(n).
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(2) = a(n).
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Plane division by lines
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A007283, A010892, A011655, A024493, A024494, A024495, A057079, A082365.
Cf. A083322, A139469, A111927, A113405, A131577, A131708, A132804, A132805.
Cf. A175805, A291000.
Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), this sequence (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2/((1-x)^3-x^3) )); // G. C. Greubel, Apr 11 2023

a:= proc(n) option remember; `if`(n=0, 0, 2*a(n-1)+
      [-1, 0, 1, 1, 0, -1, -1][1+(n mod 6)])
    end:
seq(a(n), n=0..33); # Paul Weisenhorn, May 17 2020

LinearRecurrence[{3,-3,2},{0,0,1},40] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = sum(k=0,n\3,binomial(n,3*k+2)) /* Michael Somos, Feb 14 2006 */

a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[3,1]) /* Michael Somos, Feb 14 2006 */

def A024495(n): return (2^n - chebyshev_U(n, 1/2) - chebyshev_U(n-1, 1/2))/3
[A024495(n) for n in range(41)] # G. C. Greubel, Apr 11 2023

a(n) = ( 2^n + 2*cos((n-4)*Pi/3) )/3 = (2^n - A057079(n))/3.
a(n) = 2*a(n-1) + A010892(n-2) = a(n-1) + A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - Henry Bottomley, Jun 04 2001
a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - Benoit Cloitre, Sep 04 2002
a(n) = Sum_{k=0..n} 2^k*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3) (offset 0). - Paul Barry, May 18 2004
G.f.: x^2/((1-x)^3 - x^3) =  x^2 / ( (1-2*x)*(1-x+x^2) ).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - Paul Curtz, Nov 18 2007
a(n) + A024493(n-1) = A131577(n). - Paul Curtz, Jan 24 2008
From Paul Curtz, May 29 2011: (Start)
a(n) + a(n+3) = 3*2^n = A007283(n).
a(n+6) - a(n) = 21*2^n = A175805(n).
a(n) + a(n+9) = 171*2^n.
a(n+12) - a(n) = 1365*2^n. (End)
a(n) = A113405(n) + A113405(n+1). - Paul Curtz, Jun 05 2011
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = z(n). - Stanislav Sykora, Jun 10 2012
G.f.: -x^2/( x^3 - 1 + 3*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013
a(n) = 1/18*(-4*(-1)^floor((n - 1)/3) - 6*(-1)^floor(n/3) - 3*(-1)^floor((n + 1)/3) + (-1)^(1 + floor((n + 2)/3)) + 3*2^(n + 1)). - John M. Campbell, Dec 23 2016
a(n) = (1/63)*(-40 + 21*2^n - 42*floor(n/6) + 32*floor((n+3)/6) + 16*floor((n+ 4)/6) - 24*floor((n+5)/6) - 22*floor((n+7)/6) + 21*floor((n+8)/6) + 10*floor((n+9)/6) + 5*floor((n+10)/6) + 3*floor((n+11)/6) + floor((n+ 13)/6)). - John M. Campbell, Dec 24 2016
a(n+m) = a(n)*A024493(m) + A131708(n)*A131708(m) + A024493(n)*a(m). - Vladimir Shevelev, Aug 01 2017
From Kevin Ryde, Sep 24 2020: (Start)
a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) - (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]
a(n) + A111927(n) + A131708(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1] (End)
E.g.f.: (exp(2*x) - exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Feb 06 2025

A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 1, 3, 5, 2, 1, 7, 5, 2, 1, 5, 9, 5, 2, 1, 12, 10, 5, 2, 1, 7, 17, 10, 5, 2, 1, 19, 19, 10, 5, 2, 1, 11, 28, 20, 10, 5, 2, 1, 30, 33, 20, 10, 5, 2, 1, 15, 47, 35, 20, 10, 5, 2, 1, 45, 57, 36, 20, 10, 5, 2, 1, 22, 73, 62, 36, 20, 10, 5, 2, 1

Offset: 0

Gus Wiseman, Jun 05 2021

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - Gus Wiseman, Nov 29 2021

			Triangle begins:
   1
   1
   1   1
   2   1
   2   2   1
   4   2   1
   3   5   2   1
   7   5   2   1
   5   9   5   2   1
  12  10   5   2   1
   7  17  10   5   2   1
  19  19  10   5   2   1
  11  28  20  10   5   2   1
  30  33  20  10   5   2   1
  15  47  35  20  10   5   2   1
  45  57  36  20  10   5   2   1
  22  73  62  36  20  10   5   2   1
  67  92  64  36  20  10   5   2   1
  30 114 102  65  36  20  10   5   2   1
  97 147 107  65  36  20  10   5   2   1
Row n = 10 counts the following partitions (A = 10):
  (55)          (64)         (73)       (82)     (91)   (A)
  (3322)        (442)        (433)      (622)    (811)
  (4411)        (541)        (532)      (721)
  (222211)      (3331)       (631)      (7111)
  (331111)      (4222)       (5221)     (61111)
  (22111111)    (4321)       (6211)
  (1111111111)  (5311)       (42211)
                (22222)      (52111)
                (32221)      (511111)
                (33211)      (4111111)
                (43111)
                (322111)
                (421111)
                (2221111)
                (3211111)
                (31111111)
                (211111111)
The conjugate version is:
  (A)      (55)      (3331)     (331111)    (31111111)   (1111111111)
  (64)     (73)      (5311)     (511111)    (211111111)
  (82)     (91)      (7111)     (3211111)
  (442)    (433)     (33211)    (4111111)
  (622)    (532)     (43111)    (22111111)
  (4222)   (541)     (52111)
  (22222)  (631)     (61111)
           (721)     (322111)
           (811)     (421111)
           (3322)    (2221111)
           (4321)
           (4411)
           (5221)
           (6211)
           (32221)
           (42211)
           (222211)

This is A103919 with all zeros removed.
The strict version is A152146 interleaved with A152157.
The rows are those of A239830 interleaved with those of A239829.
The reverse version is the right half of A344612.
The strict reverse version is the right half of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with rev-alternating sum <= 0, ranked by A028260.
A124754 lists alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344607 counts partitions with rev-alternating sum >= 0, ranked by A344609.
A344608 counts partitions with rev-alternating sum < 0, ranked by A119899.
A344610 counts partitions of n by positive rev-alternating sum.
A344611 counts partitions of 2n with rev-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
A346697 gives the sum of odd-indexed prime indices (reverse: A346699).
A346702 represents the odd bisection of compositions, sums A209281.
Cf. A000070, A000097, A003242, A006330, A025047, A097805, A114121, A116406, A131577, A344617, A344649, A344650, A344654, A346633.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],ats[#]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

A345197 Concatenation of square matrices A(n), each read by rows, where A(n)(k,i) is the number of compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 0, 0, 3, 4, 3, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 03 2021

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The matrices for n = 1..7:
  1   0 1   0 0 1   0 0 0 1   0 0 0 0 1   0 0 0 0 0 1   0 0 0 0 0 0 1
      1 0   1 1 0   1 1 1 0   1 1 1 1 0   1 1 1 1 1 0   1 1 1 1 1 1 0
            0 1 0   0 1 2 0   0 1 2 3 0   0 1 2 3 4 0   0 1 2 3 4 5 0
                    0 1 0 0   0 2 2 0 0   0 3 4 3 0 0   0 4 6 6 4 0 0
                              0 0 1 0 0   0 0 2 3 0 0   0 0 3 6 6 0 0
                                          0 0 1 0 0 0   0 0 3 3 0 0 0
                                                        0 0 0 1 0 0 0
Matrix n = 5 counts the following compositions:
           i=-3:        i=-1:          i=1:            i=3:        i=5:
        -----------------------------------------------------------------
   k=1: |    0            0             0               0          (5)
   k=2: |   (14)         (23)          (32)            (41)         0
   k=3: |    0          (131)       (221)(122)   (311)(113)(212)    0
   k=4: |    0       (1211)(1112)  (2111)(1121)         0           0
   k=5: |    0            0          (11111)            0           0

Gus Wiseman, A raster plot of the zeros in A(16).

The number of nonzero terms in each matrix appears to be A000096.
The number of zeros in each matrix appears to be A000124.
Row sums and column sums both appear to be A007318 (Pascal's triangle).
The matrix sums are A131577.
Antidiagonal sums appear to be A163493.
The reverse-alternating version is also A345197 (this sequence).
Antidiagonals are A345907.
Traces are A345908.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Other tetrangles: A318393, A318816, A320808, A321912.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A007318, A008549, A025047, A032443, A034871, A114121, A120452, A238279, A239830, A344604.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==k&&ats[#]==i&]],{n,0,6},{k,1,n},{i,-n+2,n,2}]

A122803 Powers of -2: a(n) = (-2)^n.

Original entry on oeis.org

1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, -2048, 4096, -8192, 16384, -32768, 65536, -131072, 262144, -524288, 1048576, -2097152, 4194304, -8388608, 16777216, -33554432, 67108864, -134217728, 268435456, -536870912, 1073741824, -2147483648, 4294967296, -8589934592, 17179869184

Offset: 0

Franklin T. Adams-Watters, Sep 11 2006

The number -2 can be used as a base of numeration (see the Weisstein link). - Alonso del Arte, Mar 30 2014
Contribution from M. F. Hasler, Oct 21 2014: (Start)
This is the inverse binomial transform of A033999 = n->(-1)^n, and the binomial transform of A033999*A000244 = n->(-3)^n, see also A141413.
Prefixed with one 0, i.e., (0,1,-2,4,...) = -A033999*A131577, it is the binomial transform of (0, 1, -4, 13, -40, 121,...) = -A033999*A003462, and inverse binomial transform of (0,1,0,1,0,1,...) = A000035.
Prefixed with two 0's, i.e., (0,0,1,-2,4,-8,...), it is the binomial transform of (0,0,1,-5,18,-58,179,-543,...) (cf. A000340) and inverse binomial transform of (0,0,1,1,2,2,3,3,...) = A004526. (End)
Prefixed with three 0's, this is the inverse binomial difference of (0, 0, 0, 1, 2, 4, 6, 9, 12, 16,...) = concat(0, A002620), which has as successive differences (0, 0, 1, 1, 2, 2,...) = A004526, then (0, 1, 0, 1,...) = A000035, then (1, -1, 1, -1,...) = A033999, and then (-2)^k*A033999 with k=1,2,3,... - Paul Curtz, Oct 16 2014, edited by M. F. Hasler, Oct 21 2014
Stirling-Bernoulli transform of triangular numbers: 1, 3, 6, 10, 15, 21, 28, ... - Philippe Deléham, May 25 2015

Franklin T. Adams-Watters, Table of n, (-2)^n for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Negabinary.
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A000079 (powers of 2), A004526, A005351, A005352, A011782, A034008, A131577, A171449, A248800.

[(-2)^n: n in [0..60]]; // Vincenzo Librandi, Oct 22 2014

A122803:=n->(-2)^n; seq(A122803(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[(-2)^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=(-2)^n \\ Charles R Greathouse IV, Sep 24 2015

def A122803(n): return -(1<Chai Wah Wu, Nov 18 2022

a(n) = (-2)^n = (-1)^n * 2^n.
a(n) = -2*a(n-1), n > 0; a(0) = 1. G.f.: 1/(1+2x). - Philippe Deléham, Nov 19 2008
Sum_{n >= 0} 1/a(n) = 2/3. - Jaume Oliver Lafont, Mar 01 2009
E.g.f.: 1/exp(2*x). - Arkadiusz Wesolowski, Aug 13 2012
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A030195(n+1). - R. J. Mathar, Oct 15 2012
G.f.: 1/(1+2x). A122803 = A033999 * A000079. - M. F. Hasler, Oct 21 2014
a(n) = Sum_{k = 0..n} A163626(n,k)*A000217(k+1). - Philippe Deléham, May 25 2015

A326675 The positions of 1's in the reversed binary expansion of n are pairwise coprime, where a singleton is not coprime unless it is {1}.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 33, 48, 49, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 96, 97, 112, 113, 129, 132, 133, 144, 145, 148, 149, 192, 193, 196, 197, 208, 209, 212

Offset: 1

Gus Wiseman, Jul 17 2019

			41 has reversed binary expansion (1,0,0,1,0,1) with positions of 1's being {1,4,6}, which are not pairwise coprime, so 41 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Equals the complement of A131577 in A087087.
Numbers whose prime indices are pairwise coprime are A302696.
Taking relatively prime instead of pairwise coprime gives A291166.
Cf. A000120, A051293, A070939, A289509, A291165, A319826, A326669, A326673, A326674.

extend:= proc(L) local C,c;
  C:= select(t -> andmap(s -> igcd(s,t)=1, L), [$1..L[-1]-1]);
  L, seq(procname([op(L),c]),c=C)
end proc:
g:= proc(L) local i;
  add(2^(i-1),i=L)
end proc:
map(g, [[1],seq(extend([k])[2..-1], k=2..10)]); # Robert Israel, Jul 19 2019

Select[Range[100],CoprimeQ@@Join@@Position[Reverse[IntegerDigits[#,2]],1]&]

is(n) = my (p=1); while (n, my (o=1+valuation(n,2)); if (gcd(p,o)>1, return (0), n-=2^(o-1); p*=o)); return (1) \\ Rémy Sigrist, Jul 19 2019

A094373 Expansion of (1-x-x^2)/((1-x)(1-2x)).

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593

Offset: 0

Paul Barry, Apr 28 2004

Partial sum of 1,1,1,2,4,8,...
Binomial transform of abs(A073097).
Binomial transform is A094374.
Partial sums are in A006127. - Paul Barry, Aug 05 2004
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the central square these vectors lead to the companion sequence A011782. - Johannes W. Meijer, Aug 15 2010
This sequence has a(0) = 1 and for all n > 0, a(n) = 2^(n-1)+1. Consequently 2*a(n) >= a(n+1) for all n > 0 and the sequence is complete. - Frank M Jackson, Jan 29 2012
Row lengths of the triangle in A198069. - Reinhard Zumkeller, May 26 2013
Take A007843 and count the repeated values. The result is 1,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,.... Build a third sequence, where a(1) = 1 and a(n) equals the length (greater than 1) of the shortest palindromic subsequence of consecutive terms of the second sequence starting with a(n) of the second sequence. The third sequence starts 1,3,5,3,9,3,5,3,17,3,5,3,9,3,5,3,33,.... Conjecturally, in the third sequence: (1) the indices of the first occurrence of each value form the present sequence and (2) for n>1, a(n) is in the a(n-1)-th position. - Ivan N. Ianakiev, Aug 20 2019

			G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 9*x^4 + 17*x^5 + 33*x^6 + 65*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein, Complete Sequence.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from the initial 1, identical to A000051.
Cf. A135225.
Column k=1 of A152977.
Row n=2 of A238016.

a:=[2,3];; for n in [3..40] do a[n]:=3*a[n-1]-2*a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Nov 06 2019

[(2^n-0^n)/2+1: n in [0..40]]; // Vincenzo Librandi, Jun 10 2011

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-x-x^2)/((1-x)*(1-2*x)))); // Marius A. Burtea, Oct 25 2019

1, seq((2^n - 0^n)/2 +1, n=1..40); # G. C. Greubel, Nov 06 2019

CoefficientList[Series[(1-x-x^2)/((1-x)*(1-2*x)), {x, 0, 40}], x] (* or *) Join[{1}, LinearRecurrence[{3, -2}, {2, 3}, 40]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)
a[ n_]:= If[n<0, 0, 1 + Quotient[2^n, 2]]; (* Michael Somos, May 26 2014 *)
a[ n_]:= SeriesCoefficient[(1-x-x^2)/((1-x)(1-2x)), {x, 0, n}]; (* Michael Somos, May 26 2014 *)
LinearRecurrence[{3,-2},{1,2,3},40] (* Harvey P. Dale, Aug 09 2015 *)

a(n)=2^n\2+1 \\ Charles R Greathouse IV, Apr 05 2013

Vec((1-x-x^2)/((1-x)*(1-2*x))+O(x^40)) \\ Charles R Greathouse IV, Apr 05 2013

[(2^n - 0^n)/2 + 1 for n in (0..40)] # G. C. Greubel, Nov 06 2019

a(n) = (2^n - 0^n)/2 + 1.
a(n) = 3*a(n-1) - 2*a(n-2).
a(2*n) = 2*a(2*n-1) - 1, n>0.
Row sums of triangle A135225. - Gary W. Adamson, Nov 23 2007
a(n) = A131577(n) + 1. - Paul Curtz, Aug 07 2008
a(n) = 2*a(n-1) - 1 for n>1, a(0)=1, a(1)=2. - Philippe Deléham, Sep 25 2009
E.g.f.: exp(x)*(1 + sinh(x)). - Arkadiusz Wesolowski, Aug 13 2012
G.f.: G(0), where G(k)= 1 + 2^k*x/(1 - x/(x + 2^k*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013
a(n) = 2^(n-1) +1 = A000051(n-1) for n>0. - M. F. Hasler, Sep 22 2013

A333766 Maximum part of the n-th composition in standard order. a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 2, 2, 1, 5, 4, 3, 3, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 6, 5, 4, 4, 3, 3, 3, 3, 4, 3, 2, 2, 3, 2, 2, 2, 5, 4, 3, 3, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 7, 6, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 5, 4, 3, 3, 3, 2, 2

Offset: 0

Gus Wiseman, Apr 05 2020

nonn

One plus the longest run of 0's in the binary expansion of n.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 100th composition in standard order is (1,3,3), so a(100) = 3.

Positions of ones are A000225.
Positions of terms <= 2 are A003754.
The version for prime indices is A061395.
Positions of terms > 1 are A062289.
Positions of first appearances are A131577.
The minimum part is given by A333768.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Max@@stc[n]],{n,0,100}]

For n > 0, a(n) = A087117(n) + 1.

A034008 a(n) = floor(2^|n-1|/2). Or: 1, 0, followed by powers of 2.

Original entry on oeis.org

1, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Wolfdieter Lang

Powers of 2 with additional first two terms.
Essentially the same as A131577 (and A000079).
[(-1)^n*a(n)] = [1, 0, 1, -2, 4, -8, 16, -32, ...] is the inverse binomial transform of A008619 = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]. - Philippe Deléham, Nov 15 2009
Number of compositions (ordered partitions) of n into an even number of parts. - Geoffrey Critzer, Mar 28 2010
Number of compositions of n into an even number of even parts.
Number of compositions of n into parts k >= 2 where there are k - 1 sorts of part k. - Joerg Arndt, Sep 30 2012
Taking n-th differences of this sequence reproduces the same sequence except for a(1) = n mod 2 (parity of n) and a(0) = (-1)^a(1)*floor(n/2 + 1). - M. F. Hasler, Jan 13 2015

Richard P. Stanley, Enumerative Combinatorics, Vol. I, Cambridge University Press, 1997, p. 45, exercise 9.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A011782, A131577, A000079.

Cf. A001057, A008619.

A034008:=n->2^(n-2): 1, 0, seq(A034008(n), n=2..50); # Wesley Ivan Hurt, Apr 12 2017

a = x/(1 - x); CoefficientList[Series[1/(1 - a^2), {x, 0, 30}], x] (* Geoffrey Critzer, Mar 28 2010 *)

PARI
```
a(n)=if(n<2,n==0,2^(n-2))
```

a(n) = 2^(n-2), n >= 2; a(0) = 1, a(1) = 0.
G.f.: (1-x)^2/(1-2*x).
G.f. 1/( 1 - Sum_{k >= 1} (k-1)*x^k ). - Joerg Arndt, Sep 30 2012
G.f.: x*G(0), where G(k) = 1 + 1/(1 - (1 - x)/(1 + x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
a(n+1) = A131577(n) and a(n+2) = A000079(n) for all n >= 0. - M. F. Hasler, Jan 13 2015
Inverse binomial transform of (3^n - 2*n + 1)/2 for n >= 0. - Paul Curtz, Sep 24 2019
E.g.f.: (1/4)*(3 + exp(2*x) - 2*x). - Stefano Spezia, Sep 25 2019
Binomial transform of A001057(n+1) or (-1)^n*A008619(n). - Paul Curtz, Oct 07 2019

Additional comments from Barry E. Williams, May 27 2000
Additional comments from Michael Somos, Jun 18 2002
Edited by M. F. Hasler, Jan 13 2015

A163493 Number of binary strings of length n which have the same number of 00 and 01 substrings.

Original entry on oeis.org

1, 2, 2, 3, 6, 9, 15, 30, 54, 97, 189, 360, 675, 1304, 2522, 4835, 9358, 18193, 35269, 68568, 133737, 260802, 509132, 995801, 1948931, 3816904, 7483636, 14683721, 28827798, 56637969, 111347879, 219019294, 431043814, 848764585, 1672056525, 3295390800, 6497536449

Offset: 0

Christopher Carl Heckman, Jul 29 2009

nonn

A variation of problem 11424 in the American Mathematical Monthly. Terms were brute-force calculated using Maple 10.
Proposed Problem 11610 in the Dec 2011 A.M.M.
From Gus Wiseman, Jul 27 2021: (Start)
Also the antidiagonal sums of the matrices counting integer compositions by length and alternating sum (A345197). So a(n) is the number of integer compositions of n + 1 of length (n - s + 3)/2, where s is the alternating sum of the composition. For example, the a(0) = 1 through a(6) = 7 compositions are:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (31)   (41)    (51)    (61)
                   (121)  (122)   (123)   (124)
                          (221)   (222)   (223)
                          (1112)  (321)   (322)
                          (1211)  (1122)  (421)
                                  (1221)  (1132)
                                  (2112)  (1231)
                                  (2211)  (2122)
                                          (2221)
                                          (3112)
                                          (3211)
                                          (11131)
                                          (12121)
                                          (13111)
For a bijection with the main (binary string) interpretation, take the run-lengths of each binary string of length n + 1 that satisfies the condition and starts with 1.
(End)

			1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 54*x^8 + 97*x^9 + ...
From _Gus Wiseman_, Jul 27 2021: (Start)
The a(0) = 1 though a(6) = 15 binary strings:
  ()  (0)  (1,0)  (0,0,1)  (0,0,1,0)  (0,0,1,1,0)  (0,0,0,1,0,1)
      (1)  (1,1)  (1,1,0)  (0,0,1,1)  (0,0,1,1,1)  (0,0,1,0,0,1)
                  (1,1,1)  (0,1,0,0)  (0,1,1,0,0)  (0,0,1,1,1,0)
                           (1,0,0,1)  (1,0,0,1,0)  (0,0,1,1,1,1)
                           (1,1,1,0)  (1,0,0,1,1)  (0,1,0,0,0,1)
                           (1,1,1,1)  (1,0,1,0,0)  (0,1,1,1,0,0)
                                      (1,1,0,0,1)  (1,0,0,1,1,0)
                                      (1,1,1,1,0)  (1,0,0,1,1,1)
                                      (1,1,1,1,1)  (1,0,1,1,0,0)
                                                   (1,1,0,0,1,0)
                                                   (1,1,0,0,1,1)
                                                   (1,1,0,1,0,0)
                                                   (1,1,1,0,0,1)
                                                   (1,1,1,1,1,0)
                                                   (1,1,1,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3328 (first 501 terms from R. H. Hardin)
Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From _N. J. A. Sloane_, Apr 07 2012]
R. Stanley, Problem 11610, Amer. Math. Monthly, 118 (2011), 937; 120 (2013), 943-944.

Antidiagonal sums of the matrices A345197.
Row sums of A345907.
Taking diagonal instead of antidiagonal sums gives A345908.
A011782 counts compositions (or binary strings).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000041, A000070, A000096, A000097, A000124, A000346, A007318, A008549, A025047, A131577, A238279.

with(combinat): count := proc(n) local S, matches, A, k, i; S := subsets(\{seq(i, i=1..n)\}): matches := 0: while not S[finished] do A := S[nextvalue](): k := 0: for i from 1 to n-1 do: if not (i in A) and not (i+1 in A) then k := k + 1: fi: if not (i in A) and (i+1 in A) then k := k - 1: fi: od: if (k = 0) then matches := matches + 1: fi: end do; return(matches); end proc:
# second Maple program:
b:= proc(n, l, t) option remember; `if`(n-abs(t)<0, 0, `if`(n=0, 1,
      add(b(n-1, i, t+`if`(l=0, (-1)^i, 0)), i=0..1)))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..36);  # Alois P. Heinz, Mar 20 2024

a[0] = 1; a[n_] := Sum[Binomial[2*k - 1, k]*Binomial[n - 2*k, k] + Binomial[2*k, k]*Binomial[n - 2*k - 1, k], {k, 0, n/3}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 28 2017, after Joel B. Lewis *)
Table[Length[Select[Tuples[{0,1},n],Count[Partition[#,2,1],{0,0}]==Count[Partition[#,2,1],{0,1}]&]],{n,0,10}] (* Gus Wiseman, Jul 27 2021 *)
a[0]:=1; a[n_]:=(1 + 3*HypergeometricPFQ[{1/2, 1-3*n/8, (1-n)/3, (2-n)/3, -n/3},{1, (1-n)/2, 1-n/2, -3*n/8}, -27])/2; Array[a,37,0] (* Stefano Spezia, Apr 26 2024 *)

from math import comb
def A163493(n): return 2+sum((x:=comb((k:=m<<1)-1,m)*comb(n-k,m))+(x*(n-3*m)<<1)//(n-k) for m in range(1,n//3+1)) if n else 1 # Chai Wah Wu, May 01 2024

G.f.: 1/2/(1-x) + (1+2*x)/2/sqrt((1-x)*(1-2*x)*(1+x+2*x^2)). - Richard Stanley, corrected Apr 29 2011
G.f.: (1 + sqrt( 1 + 4*x / ((1 - x) * (1 - 2*x) * (1 + x + 2*x^2)))) / (2*(1 - x)). - Michael Somos, Jan 30 2012
a(n) = sum( binomial(2*k-1, k)*binomial(n-2*k,k) + binomial(2*k, k)*binomial(n-2*k-1, k), k=0..floor(n/3)). - Joel B. Lewis, May 21 2011
Conjecture: -n*a(n) +(2+n)*a(n-1) +(3n-12)*a(n-2) +(12-n)*a(n-3) +(2n-18)*a(n-4)+(56-12n)*a(n-5) +(8n-40)*a(n-6)=0. - R. J. Mathar, Nov 28 2011
G.f. y = A(x) satisfies x = (1 - x) * (1 - 2*x) * (1 + x + 2*x^2) * y * (y * (1 - x) - 1). - Michael Somos, Jan 30 2012
Sequence a(n) satisfies 0 = a(n) * (n^2-2*n) + a(n-1) * (-3*n^2+8*n-2) + a(n-2) * (3*n^2-10*n+2) + a(n-3) * (-5*n^2+18*n-6) + a(n-4) * (8*n^2-34*n+22) + a(n-5) * (-4*n^2+20*n-16) except if n=1 or n=2. - Michael Somos, Jan 30 2012
a(n) = (1 + 3*hypergeom([1/2, 1-3*n/8, (1-n)/3, (2-n)/3, -n/3],[1, (1-n)/2, 1-n/2, -3*n/8],-27))/2 for n > 0. - Stefano Spezia, Apr 26 2024
a(n) ~ 2^n / sqrt(Pi*n). - Vaclav Kotesovec, Apr 26 2024

cp's OEIS Frontend

A005418 Number of (n-1)-bead black-white reversible strings; also binary grids; also row sums of Losanitsch's triangle A034851; also number of caterpillar graphs on n+2 vertices.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A345197 Concatenation of square matrices A(n), each read by rows, where A(n)(k,i) is the number of compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A122803 Powers of -2: a(n) = (-2)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A326675 The positions of 1's in the reversed binary expansion of n are pairwise coprime, where a singleton is not coprime unless it is {1}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A094373 Expansion of (1-x-x^2)/((1-x)*(1-2*x)).

Views

Author

A094373 Expansion of (1-x-x^2)/((1-x)(1-2x)).