cp's OEIS Frontend

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.

Showing 1-9 of 9 results.

A001405 a(n) = binomial(n, floor(n/2)).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 705432, 1352078, 2704156, 5200300, 10400600, 20058300, 40116600, 77558760, 155117520, 300540195, 601080390, 1166803110
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Sperner's theorem says that this is the maximal number of subsets of an n-set such that no one contains another.
When computed from index -1, [seq(binomial(n,floor(n/2)), n = -1..30)]; -> [1,1,1,2,3,6,10,20,35,70,126,...] and convolved with aerated Catalan numbers [seq(((n+1) mod 2)*binomial(n,n/2)/((n/2)+1), n = 0..30)]; -> [1,0,1,0,2,0,5,0,14,0,42,0,132,0,...] shifts left by one: [1,1,2,3,6,10,20,35,70,126,252,...] and if again convolved with aerated Catalan numbers, gives A037952 apart from the initial term. - Antti Karttunen, Jun 05 2001 [This is correct because the g.f.'s satisfy (1+x*g001405(x))*g126120(x) = g001405(x) and g001405(x)*g126120(x) = g037952(x)/x. - R. J. Mathar, Sep 23 2021]
Number of ordered trees with n+1 edges, having nonroot nodes of outdegree 0 or 2. - Emeric Deutsch, Aug 02 2002
Gives for n >= 1 the maximum absolute column sum norm of the inverse of the Vandermonde matrix (a_ij) i=0..n-1, j=0..n-1 with a_00=1 and a_ij=i^j for (i,j) != (0,0). - Torsten Muetze, Feb 06 2004
Image of Catalan numbers A000108 under the Riordan array (1/(1-2x),-x/(1-2x)) or A065109. - Paul Barry, Jan 27 2005
Number of left factors of Dyck paths, consisting of n steps. Example: a(4)=6 because we have UDUD, UDUU, UUDD, UUDU, UUUD and UUUU, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Apr 23 2005
Number of dispersed Dyck paths of length n; they are defined as concatenations of Dyck paths and (1,0)-steps on the x-axis; equivalently, Motzkin paths with no (1,0)-steps at positive height. Example: a(4)=6 because we have HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD, where U=(1,1), H=(1,0), and D=(1,-1). - Emeric Deutsch, Jun 04 2011
a(n) is odd iff n=2^k-1. - Jon Perry, May 05 2005
An inverse Chebyshev transform of binomial(1,n)=(1,1,0,0,0,...) where g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), with c(x) the g.f. of A000108. - Paul Barry, May 13 2005
In a random walk on the number line, starting at 0 and with 0 absorbing after the first step, number of ways of ending up at a positive integer after n steps. - Joshua Zucker, Jul 31 2005
Maximum number of sums of the form Sum_{i=1..n} e(i)*a(i) that are congruent to 0 mod q, where e_i=0 or 1 and gcd(a_i,q)=1, provided that q > ceiling(n/2). - Ralf Stephan, Apr 27 2003
Also the number of standard tableaux of height <= 2. - Mike Zabrocki, Mar 24 2007
Hankel transform of this sequence forms A000012 = [1,1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
A001263 * [1, -2, 3, -4, 5, ...] = [1, -1, -2, 3, 6, -10, -20, 35, 70, -126, ...]. - Gary W. Adamson, Jan 02 2008
Equals right border of triangle A153585. - Gary W. Adamson, Dec 28 2008
Second binomial transform of A168491. - Philippe Deléham, Nov 27 2009
a(n) is also the number of distinct strings of length n, each of which is a prefix of a string of balanced parentheses; see example. - Lee A. Newberg, Apr 26 2010
Number of symmetric balanced strings of n pairs of parentheses; see example. - Joerg Arndt, Jul 25 2011
a(n) is the number of permutation patterns modulo 2. - Olivier Gérard, Feb 25 2011
For n >= 2, a(n-1) is the number of incongruent two-color bracelets of 2*n-1 beads, n of which are black (A007123), having a diameter of symmetry. - Vladimir Shevelev, May 03 2011
The number of permutations of n elements where p(k-2) < p(k) for all k. - Joerg Arndt, Jul 23 2011
Also size of the equivalence class of S_{n+1} containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> cba where a < b < c, cf. A210668. - Tom Roby, May 15 2012
a(n) is the number of symmetric Dyck paths of length 2n. - Matt Watson, Sep 26 2012
a(n) is divisible by A000108(floor(n/2)) = abs(A129996(n-2)). - Paul Curtz, Oct 23 2012
a(n) is the number of permutations of length n avoiding both 213 and 231 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Number of symmetric standard Young tableaux of shape (n,n). - Ran Pan, Apr 10 2015
From Luciano Ancora, May 09 2015: (Start)
Also "stepped path" in the array formed by partial sums of the all 1's sequence (or a Pascal's triangle displayed as a square). Example:
[1], [1], 1, 1, 1, 1, 1, ... A000012
1, [2], [3], 4, 5, 6, 7, ...
1, 3, [6], [10], 15, 21, 28, ...
1, 4, 10, [20], [35], 56, 84, ...
1, 5, 15, 35, [70], [126], 210, ...
Sequences in second formula are the mixed diagonals shown in this array. (End)
a(n) = A265848(n,n). - Reinhard Zumkeller, Dec 24 2015
The constant Sum_{n >= 0} a(n)/n! is 1 + A130820. - Peter Bala, Jul 02 2016
Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-1,1}. - David Nguyen, Dec 20 2016
a(n) is also the number of paths of n steps (either up or down by 1) that end at the maximal value achieved along the path. - Winston Luo, Jun 01 2017
Number of binary n-tuples such that the number of 1's in the even positions is the same as the number of 1's in the odd positions. - Juan A. Olmos, Dec 21 2017
Equivalently, a(n) is the number of subsets of {1,...,n} containing as many even numbers as odd numbers. - Gus Wiseman, Mar 17 2018
a(n) is the number of Dyck paths with semilength = n+1, returns to the x-axis = floor((n+3)/2) and up movements in odd positions = floor((n+3)/2). Example: a(4)=6, U=up movement in odd position, u=up movement in even position, d=down movement, -=return to x-axis: Uududd-Ud-Ud-, Ud-Uudd-Uudd-, Uudd-Uudd-Ud-, Ud-Ud-Uududd-, Uudd-Ud-Uudd-, Ud-Uududd-Ud-. - Roger Ford, Dec 29 2017
Let C_n(R, H) denote the transition matrix from the ribbon basis to the homogeneous basis of the graded component of the algebra of noncommutative symmetric functions of order n. Letting I(2^(n-1)) denote the identity matrix of order 2^(n-1), it has been conjectured that the dimension of the kernel of C_n(R, H) - I(2^(n-1)) is always equal to a(n-1). - John M. Campbell, Mar 30 2018
The number of U-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are U-equivalent iff the positions of pattern U are identical in these paths. - Sergey Kirgizov, Apr 08 2018
All binary self-dual codes of length 2n, for n > 0, must contain at least a(n) codewords of weight n. More to the point, there will always be at least one, perhaps unique, binary self-dual code of length 2n that will contain exactly a(n) codewords that have a hamming weight equal to half the length of the code (n). This code can be constructed by direct summing the unique binary self-dual code of length 2 (up to permutation equivalence) to itself n times. A permutation equivalent code can be constructed by augmenting two identity matrices of length n together. - Nathan J. Russell, Nov 25 2018
Closed under addition. - Torlach Rush, Apr 18 2019
The sequence starting (1, 2, 3, 6, ...) is the invert transform of A097331: (1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, ...). - Gary W. Adamson, Feb 22 2020
From Gary W. Adamson, Feb 24 2020: (Start)
The sequence is the culminating limit of an infinite set of sequences with convergents of 2*cos(Pi/N), N = (3, 5, 7, 9, ...).
The first few such sequences are:
N = 3: (1, 1, 1, 1, 1, 1, 1, 1, ...)
N = 5: (1, 1, 2, 3, 5, 8, 13, 21, ...) = A000045
N = 7: (1, 1, 2, 3, 6, 10, 19, 33, ...) = A028495, a(n)/a(n-1) tends to 1.801937...
N = 9 (1, 1, 2, 3, 6, 10, 20, 35, ...) = A061551, a(n)/a(n_1) tends to 1.879385...
...
In the limit one gets the current sequence with ratio 2. (End)
a(n) is also the number of monotone lattice paths from (0,0) to (floor(n/2),ceiling(n/2)). These are the number of Grand Dyck paths when n is even. - Nachum Dershowitz, Aug 12 2020
The maximum number of preimages that a permutation of length n+1 can have under the consecutive-132-avoiding stack-sorting map. - Colin Defant, Aug 28 2020
Counts faro permutations of length n. Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one. - Sergey Kirgizov, Jan 12 2021
Per "Sperner's Theorem", the largest possible familes of finite sets none of which contain any other sets in the family. - Renzo Benedetti, May 26 2021
a(n-1) are the incomplete, primitive Dyck paths of n steps without a first return: paths of U and D steps starting at the origin, never touching the horizontal axis later on, and ending above the horizontal axis. n=1: {U}, n=2: {UU}, n=3: {UUU, UUD}, n=4: {UUUU, UUUD, UUDU}, n=5: {UUUUU, UUUUD, UUUDD, UUDUU, UUUDU, UUDUD}. For comparison: A037952 counts incomplete Dyck paths with n steps with any number of intermediate returns to the horizontal axis, ending above the horizontal axis. - R. J. Mathar, Sep 24 2021
a(n) is the number of noncrossing partitions of [n] whose nontrivial blocks are of type {a,b}, with a <= n/2, b > n/2. - Francesca Aicardi, May 29 2022
Maximal coefficient of (1+x)^n. - Vaclav Kotesovec, Dec 30 2022
Sums of lower-left-to-upper-right diagonals of the Catalan Triangle A001263. - Howard A. Landman, Sep 16 2024

Examples

			For n = 4, the a(4) = 6 distinct strings of length 4, each of which is a prefix of a string of balanced parentheses, are ((((, (((), (()(, ()((, ()(), and (()). - _Lee A. Newberg_, Apr 26 2010
There are a(5)=10 symmetric balanced strings of 5 pairs of parentheses:
[ 1] ((((()))))
[ 2] (((()())))
[ 3] ((()()()))
[ 4] ((())(()))
[ 5] (()()()())
[ 6] (()(())())
[ 7] (())()(())
[ 8] ()()()()()
[ 9] ()((()))()
[10] ()(()())() - _Joerg Arndt_, Jul 25 2011
G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 35*x^7 + 70*x^8 + ...
The a(4)=6 binary 4-tuples such that the number of 1's in the even positions is the same as the number of 1's in the odd positions are 0000, 1100, 1001, 0110, 0011, 1111. - _Juan A. Olmos_, Dec 21 2017

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 135.
  • K. Engel, Sperner Theory, Camb. Univ. Press, 1997; Theorem 1.1.1.
  • P. Frankl, Extremal sets systems, Chap. 24 of R. L. Graham et al., eds, Handbook of Combinatorics, North-Holland.
  • J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
  • 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).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), p. 452.

Links

Crossrefs

Row sums of Catalan triangle A053121 and of symmetric Dyck paths A088855.
Enumerates the structures encoded by A061854 and A061855.
First differences are in A037952.
Apparently a(n) = lim_{k->infinity} A094718(k, n).
Partial sums are in A036256. Column k=2 of A182172. Column k=1 of A335570.
Bisections: A000984 (even part), A001700 (odd part).
Cf. A000712, A001006, A001700, A005773, A005817, A007578, A007579, A022916, A022917 (permutation patterns mod k), A049401, A051920, A063886, A130820, A132815, A153585, A239241, A265848.
Cf. A097331.
Cf. A000045, A028495, A061551
Cf. A107373, A340567, A340568, A340569 (popularity of certain patterns in faro permutations).

Programs

  • GAP
    List([0..40],n->Binomial(n,Int(n/2))); # Muniru A Asiru, Apr 08 2018
  • Haskell
    a001405 n = a007318_row n !! (n `div` 2) -- Reinhard Zumkeller, Nov 09 2011
  • Magma
    [Binomial(n, Floor(n/2)): n in [0..40]]; // Vincenzo Librandi, Nov 16 2014
  • Maple
    A001405 := n->binomial(n, floor(n/2)): seq(A001405(n), n=0..33);
  • Mathematica
    Table[Binomial[n, Floor[n/2]], {n, 0, 40}] (* Stefan Steinerberger, Apr 08 2006 *)
Table[DifferenceRoot[Function[{a,n},{-4 n a[n]-2 a[1+n]+(2+n) a[2+n] == 0,a[1] == 1,a[2] == 1}]][n], {n, 30}] (* Luciano Ancora, Jul 08 2015 *)
Array[Binomial[#,Floor[#/2]]&,40,0] (* Harvey P. Dale, Mar 05 2018 *)
  • Maxima
    A001405(n):=binomial(n,floor(n/2))$
makelist(A001405(n),n,0,30); /* Martin Ettl, Nov 01 2012 */
  • PARI
    a(n) = binomial(n, n\2);
  • PARI
    first(n) = x='x+O('x^n); Vec((-1+2*x+sqrt(1-4*x^2))/(2*x-4*x^2)) \\ Iain Fox, Dec 20 2017 (edited by Iain Fox, May 07 2018)
  • Python
    from math import comb
def A001405(n): return comb(n,n//2) # Chai Wah Wu, Jun 07 2022

Formula

a(n) = max_{k=0..n} binomial(n, k).
a(2*n) = A000984(n), a(2*n+1) = A001700(n).
By symmetry, a(n) = binomial(n, ceiling(n/2)). - Labos Elemer, Mar 20 2003
P-recursive with recurrence: a(0) = 1, a(1) = 1, and for n >= 2, (n+1)*a(n) = 2*a(n-1) + 4*(n-1)*a(n-2). - Peter Bala, Feb 28 2011
G.f.: (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1 - x - x^2*c(x^2)); where c(x) = g.f. for Catalan numbers A000108.
G.f.: (-1 + 2*x + sqrt(1-4*x^2))/(2*x - 4*x^2). - Lee A. Newberg, Apr 26 2010
G.f.: 1/(1 - x - x^2/(1 - x^2/(1 - x^2/(1 - x^2/(1 - ... (continued fraction). - Paul Barry, Aug 12 2009
a(0) = 1; a(2*m+2) = 2*a(2*m+1); a(2*m+1) = Sum_{k = 0..2*m} (-1)^k*a(k)*a(2*m-k). - Len Smiley, Dec 09 2001
G.f.: (sqrt((1+2*x)/(1-2*x)) - 1)/(2*x). - Vladeta Jovovic, Apr 28 2003
The o.g.f. A(x) satisfies A(x) + x*A^2(x) = 1/(1-2*x). - Peter Bala, Feb 28 2011
E.g.f.: BesselI(0, 2*x) + BesselI(1, 2*x). - Vladeta Jovovic, Apr 28 2003
a(0) = 1; a(2*m+2) = 2*a(2*m+1); a(2*m+1) = 2*a(2*m) - c(m), where c(m)=A000108(m) are the Catalan numbers. - Christopher Hanusa (chanusa(AT)washington.edu), Nov 25 2003
a(n) = Sum_{k=0..n} (-1)^k*2^(n-k)*binomial(n, k)*A000108(k). - Paul Barry, Jan 27 2005
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(1, n-2*k). - Paul Barry, May 13 2005
From Paul Barry, Nov 02 2004: (Start)
a(n) = Sum_{k=0..floor((n+1)/2)} (binomial(n+1, k)*(cos((n-2*k+1)*Pi/2) + sin((n-2*k+1)*Pi/2))).
a(n) = Sum_{k=0..n+1}, (binomial(n+1, (n-k+1)/2)*(1-(-1)^(n-k))*(cos(k*Pi/2) + sin(k*Pi))/2). (End)
a(n) = Sum_{k=floor(n/2)..n} (binomial(n,n-k) - binomial(n,n-k-1)). - Paul Barry, Sep 06 2007
Inverse binomial transform of A005773 starting (1, 2, 5, 13, 35, 96, ...) and double inverse binomial transform of A001700. Row sums of triangle A132815. - Gary W. Adamson, Aug 31 2007
a(n) = Sum_{k=0..n} A120730(n,k). - Philippe Deléham, Oct 16 2008
a(n) = Sum_{k = 0..floor(n/2)} (binomial(n,k) - binomial(n,k-1)). - Nishant Doshi (doshinikki2004(AT)gmail.com), Apr 06 2009
Sum_{n>=0} a(n)/10^(n+1) = 0.1123724... = (sqrt(3)-sqrt(2))/(2*sqrt(2)); Sum_{n>=0} a(n)/100^(n+1) = 0.0101020306102035... = (sqrt(51)-sqrt(49))/(2*sqrt(49)). - Mark Dols, Jul 15 2010
Conjectured: a(n) = 2^n*2F1(1/2,-n;2;2), useful for number of paths in 1-d for which the coordinate is never negative. - Benjamin Phillabaum, Feb 20 2011
a(2*m+1) = (2*m+1)*a(2*m)/(m+1), e.g., a(7) = (7/4)*a(6) = (7/4)*20 = 35. - Jon Perry, Jan 20 2011
From Peter Bala, Feb 28 2011: (Start)
Let F(x) be the logarithmic derivative of the o.g.f. A(x). Then 1+x*F(x) is the o.g.f. for A027306.
Let G(x) be the logarithmic derivative of 1+x*A(x). Then x*G(x) is the o.g.f. for A058622. (End)
Let M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and [1,0,0,0,...] in the main diagonal; and V = the vector [1,0,0,0,...]. a(n) = M^n*V, leftmost term. - Gary W. Adamson, Jun 13 2011
Let M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and [1,0,0,0,...] in the main diagonal. a(n) = M^n_{1,1}. - Corrected by Gary W. Adamson, Jan 30 2012
a(n) = A007318(n, floor(n/2)). - Reinhard Zumkeller, Nov 09 2011
a(n+1) = Sum_{k=0..n} a(n-k)*A097331(k) = a(n) + Sum_{k=0..(n-1)/2} A000108(k)*a(n-2*k-1). - Philippe Deléham, Nov 27 2011
a(n) = A214282(n) - A214283(n), for n > 0. - Reinhard Zumkeller, Jul 14 2012
a(n) = Sum_{k=0..n} A168511(n,k)*(-1)^(n-k). - Philippe Deléham, Mar 19 2013
a(n+2*p-2) = Sum_{k=0..floor(n/2)} A009766(n-k+p-1, k+p-1) + binomial(n+2*p-2, p-2), for p >= 1. - Johannes W. Meijer, Aug 02 2013
O.g.f.: (1-x*c(x^2))/(1-2*x), with the o.g.f. c(x) of Catalan numbers A000108. See the rewritten formula given by Lee A. Newberg above. This is the o.g.f. for the row sums the Riordan triangle A053121. - Wolfdieter Lang, Sep 22 2013
a(n) ~ 2^n / sqrt(Pi * n/2). - Charles R Greathouse IV, Oct 23 2015
a(n) = 2^n*hypergeom([1/2,-n], [2], 2). - Vladimir Reshetnikov, Nov 02 2015
a(2*k) = Sum_{i=0..k} binomial(k, i)*binomial(k, i), a(2*k+1) = Sum_{i=0..k} binomial(k+1, i)*binomial(k, i). - Juan A. Olmos, Dec 21 2017
a(0) = 1, a(n) = 2 * a(n-1) for even n, a(n) = (2*n/(n+1)) * a(n-1) for odd n. - James East, Sep 25 2019
a(n) = A037952(n) + A000108(n/2) where A(.)=0 for non-integer argument. - R. J. Mathar, Sep 23 2021
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 2*Pi/(3*sqrt(3)) + 2.
Sum_{n>=0} (-1)^n/a(n) = 2/3 - 2*Pi/(9*sqrt(3)). (End)
For k>2, Sum_{n>=0} a(n)/k^n = (sqrt((k+2)/(k-2)) - 1)*k/2. - Vaclav Kotesovec, May 13 2022
From Peter Bala, Mar 24 2023: (Start)
a(n) = Sum_{k = 0..n+1} (-1)^(k+binomial(n+2,2)) * k/(n+1) * binomial(n+1,k)^2.
(n + 1)*(2*n - 1)*a(n) = (-1)^(n+1)*2*a(n-1) + 4*(n - 1)*(2*n + 1)*a(n-2) with a(0) = a(1) = 1. (End)
a(n) = Integral_{x=-2..2} x^n*W(x)*dx, n>=0, where W(x) = sqrt((2+x)/(2-x))/(2*Pi) is a positive function on x=(-2,2) and is singular at x = 2. Therefore a(n) is a positive definite sequence. - Karol A. Penson, May 12 2025

A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

0, 2, 10, 12, 42, 44, 50, 52, 56, 170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240, 682, 684, 690, 692, 696, 714, 716, 722, 724, 728, 738, 740, 744, 752, 810, 812, 818, 820, 824, 842, 844, 850, 852, 856, 866, 868, 872, 880, 906, 908, 914
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Comments

The binary Dyck-Language (A063171) in decimal representation.
These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, non-crossing handshakes and partitions and many other combinatorial structures in Catalan family, enumerated by A000108.
Is Sum_{k=1..n} a(k) / n^(5/2) bounded? - Benoit Cloitre, Aug 18 2002
This list is the intersection of A061854 and A031443. - Jason Kimberley, Jan 18 2013
The sequence does start at n = 0, since in the binary interpretation of the Dyck language (e.g., as parenthesizations where "1" stands for "(" and "0" stands for ")") having a(0) = 0 will do since it would stand for the empty string where the "0"s and "1"s are balanced (hence the parentheses are balanced). - Daniel Forgues, Feb 17 2013
It appears that for n>=1 this sequence can be obtained by concatenating the terms of the irregular array whose n-th row length is A000108(n) and that is defined recursively by B(n,0) = A020988(n) and B(n,k) = B(n, k-1) + D(n, k-1) where D(x,y) = (2^(2*(A089309(B(x,y))-1))-1)*(2/3) + 2^A007814(B(x,y)). - Raúl Mario Torres Silva and Michel Marcus, May 01 2020
This encoding is related to the ranking by standard ordered tree numbers in that (1) the binary encoding of trees ordered by standard ranking is given by A358505, while (2) the standard ranking of trees ordered by binary encoding is given by A358523. - Gus Wiseman, Nov 21 2022

Examples

			a(19) = 226_10 = 11100010_2 = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depth-first fashion, with 1's in binary expansion standing for internal (branching) nodes and 0's for leaves:
  0   0
   \ /
    1   0 0  (0)
     \ /   \ /
      1     1
       \   /
         1
Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular S-expression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car" and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e., the same bracket expression as above, but surrounded by extra parentheses. This mapping is performed by the Scheme function A014486->parenthesization given below.
From _Gus Wiseman_, Nov 21 2022: (Start)
The terms and corresponding ordered rooted trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))
(End)

References

  • Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 443 (Algorithm P).

Links

Crossrefs

Characteristic function: A080116. Inverse function: A080300.
The terms of binary width 2n are counted by A000108(n). Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.
Cf. also A009766, A014137, A071156, A079436, A085184, A213704.
Cf. A020988, A089309, A007814.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
The standard ranking of the ordered tree is A358523.
The depth of the ordered tree is A358550.
Nodes of the ordered tree are counted by A358551.
Cf. A000081, A001263, A358505, A358524.

Programs

  • Maple
    # Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book and the Scheme function CatalanUnrank from Ruskey's thesis. See the a089408.c program for the corresponding C procedures.
CatalanSequences := proc(upto_n) local n,a,r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n,n)/(n+1))-1 do a := [op(a),CatalanUnrank(n,r)]; od; od; return a; end;
CatalanUnrank := proc(n,rr) local r,x,y,lo,m,a; r := (binomial(2*n,n)/(n+1))-(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n,x,y+1); if(r <= lo+m-1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y-1; a := 2*a; fi; od; return a; end;
Mn := (n,x,y) -> binomial(2*n-x,n-((x+y)/2)) - binomial(2*n-x,n-1-((x+y)/2));
# Alternative:
bin := n -> ListTools:-Reverse(convert(n, base, 2)):
isA014486 := proc(n): local B, s, b; s := 0;
    if n > 0 then
      for b in bin(n) do
          s := s + ifelse(b = 1, 1, -1);
           if 0 > s then return false fi;
      od fi;
  s = 0 end:
select(isA014486, [seq(0..240)]);  # Peter Luschny, Mar 13 2024
  • Mathematica
    cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]
d2b[n_Integer] := IntegerDigits[n, 2]
tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]
nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]
wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]-1 ]
Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten
(* Alternative code *)
tbQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Count[idn2,1]==Length[idn2]/2&&Min[Accumulate[idn2/.{0->-1}]]>=0]; Join[{0},Select[Range[900],tbQ]] (* Harvey P. Dale, Jul 04 2013 *)
balancedQ[0] = True; balancedQ[n_] := Module[{s = 0}, Do[s += If[b == 1, 1, -1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0] ]; A014486 = FromDigits /@ IntegerDigits[Select[Range[0, 1000], balancedQ ]] (* Jean-François Alcover, Mar 05 2016 *)
A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; Select[Range[0, 880], A014486Q] (* JungHwan Min, Dec 11 2016 *)
(* Uses Algorithm P from Knuth's TAOCP section 7.2.1.6 - see References and Links. *)
alist[n_] := Block[{a = Flatten[Table[{1, 0}, n]], m = 2*n - 1, j, k},
    FromDigits[#, 2]& /@ Reap[
    While[True,
        Sow[a]; a[[m]] = 0;
        If[a[[m - 1]] == 0,
            a[[--m]] = 1, j = m - 1; k = 2*n - 1;
            While[j > 1 && a[[j]] == 1, a[[j--]] = 0; a[[k]] = 1; k -= 2];
            If[j == 1, Break[]];
            a[[j]] = 1; m = 2*n - 1]
    ]][[2, 1]]];
Join[{{0}, {2}}, Array[alist, 4, 2]] (* Paolo Xausa, Mar 16 2024 *)
  • PARI
    isA014486(n)=my(v=binary(n),t=0);for(i=1,#v,t+=if(v[i],1,-1);if(t<0,return(0))); t==0 \\ Charles R Greathouse IV, Jun 10 2011
  • PARI
    a_rows(N) = my(a=Vec([[0]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, my(v=if(t, valuation(t, 2), 0)); for(i=0, v, listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 16 2024
  • Python
    from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A014486_gen(): # generator of terms
    yield 0
    for l in count(1):
        for s in multiset_permutations('0'*l+'1'*(l-1)):
            c, m = 0, (l<<1)-1
            for i in range(m):
                if s[i] == '1':
                    c += 2
                if cA014486_list = list(islice(A014486_gen(),30)) # Chai Wah Wu, Nov 28 2023
  • SageMath
    def is_A014486(n) :
    B = bin(n)[2::] if n != 0 else 0
    s = 0
    for b in B :
        s += 1 if b=='1' else -1
        if 0 > s : return False
    return 0 == s
def A014486_list(n): return [k for k in (1..n) if is_A014486(k) ]
A014486_list(888) # Peter Luschny, Aug 10 2012

Extensions

Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004
Added a(0)=0 (which had been removed in June 2011), Joerg Arndt, Feb 27 2013

A031443 Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.

Original entry on oeis.org

2, 9, 10, 12, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228, 232, 240, 527, 535, 539, 541, 542, 551
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Also numbers k such that the binary digital mean dm(2, k) = (Sum_{i=1..d} 2*d_i - 1) / (2*d) = 0, where d is the number of digits in the binary representation of k and d_i the individual digits. - Reikku Kulon, Sep 21 2008
From Reikku Kulon, Sep 29 2008: (Start)
Each run of values begins with 2^(2k + 1) + 2^(k + 1) - 2^k - 1. The initial values increase according to the sequence {2^(k - 1), 2^(k - 2), 2^(k - 3), ..., 2^(k - k)}.
After this, the values follow a periodic sequence of increases by successive powers of two with single odd values interspersed.
Each run ends with an odd increase followed by increases of {2^(k - k), ..., 2^(k - 2), 2^(k - 1), 2^k}, finally reaching 2^(2k + 2) - 2^(k + 1).
Similar behavior occurs in other bases. (End)
Numbers k such that A000120(k)/A070939(k) = 1/2. - Ctibor O. Zizka, Oct 15 2008
Subsequence of A053754; A179888 is a subsequence. - Reinhard Zumkeller, Jul 31 2010
A000120(a(n)) = A023416(a(n)); A037861(a(n)) = 0.
A001700 gives number of terms having length 2*n in binary representation: A001700(n-1) = #{m: A070939(a(m))=2*n}. - Reinhard Zumkeller, Jun 08 2011
The number of terms below 2^k is A079309(floor(k/2)) for k > 1. - Amiram Eldar, Nov 21 2020

Examples

			9 is a term because '1001' contains 2 '0's and 2 '1's.

Links

Crossrefs

Subsequences: A144798, A144799, A144800, A144801, A144812, A179888.
Subsequence of A053754.
Row n = 2 of A378000.
Cf. A049354-A049360, A000120, A001316, A006519, A023416, A070939, A144777, A145057, A145058, A145059, A145060, A144912, A145037, A191292, A090050, A014486, A061854, A037861, A079309.
Terms of binary width n are enumerated by A001700.

Programs

  • Haskell
    -- See link, showing that Ulrich Schimkes formula provides a very efficient algorithm. Reinhard Zumkeller, Jun 15 2011
  • Magma
    [ n: n in [2..250] | Multiplicity({* z: z in Intseq(n,2) *}, 0) eq &+Intseq(n,2) ];  // Bruno Berselli, Jun 07 2011
  • Maple
    a:=proc(n) local nn, n1, n0: nn:=convert(n,base,2): n1:=add(nn[i],i=1..nops(nn)): n0:=nops(nn)-n1: if n0=n1 then n else end if end proc: seq(a(n), n = 1..240); # Emeric Deutsch, Jul 31 2008
  • Mathematica
    Select[Range[250],DigitCount[#,2,1]==DigitCount[#,2,0]&] (* Harvey P. Dale, Jul 22 2013 *)
FromDigits[#,2]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},2n,{1,0}],{n,5}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)
  • PARI
    for(n=1,100,b=binary(n); l=length(b); if(sum(i=1,l, component(b,i))==l/2,print1(n,",")))
  • PARI
    is(n)=hammingweight(n)==hammingweight(bitneg(n,#binary(n))) \\ Charles R Greathouse IV, Mar 29 2013
  • PARI
    is(n)=2*hammingweight(n)==exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020
  • Perl
    for my $half ( 1 .. 4 ) {
  my $N = 2 * $half;  # only even widths apply
  my $vector = (1 << ($N-1)) | ((1 << ($N/2-1)) - 1);  # first key
  my $n = 1; $n *= $_ for 2 .. $N;    # N!
  my $d = 1; $d *= $_ for 2 .. $N/2;  # (N/2)!
  for (1 .. $n/($d*$d*2)) {
    print "$vector, ";
    my ($v, $d) = ($vector, 0);
    until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }
    $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key
  }
} # Ruud H.G. van Tol, Mar 30 2014
  • Python
    from sympy.utilities.iterables import multiset_permutations
A031443_list = [int('1'+''.join(p),2) for n in range(1,10) for p in multiset_permutations('0'*n+'1'*(n-1))] # Chai Wah Wu, Nov 15 2019

Formula

a(n+1) = a(n) + 2^k + 2^(m-1) - 1 + floor((2^(k+m) - 2^k)/a(n))*(2^(2*m) + 2^(m-1)) where k is the largest integer such that 2^k divides a(n) and m is the largest integer such that 2^m divides a(n)/2^k+1. - Ulrich Schimke (UlrSchimke(AT)aol.com)
A145037(a(n)) = 0. - Reikku Kulon, Oct 02 2008

A049354 Digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's.

Original entry on oeis.org

11, 15, 19, 21, 260, 266, 268, 278, 290, 294, 302, 304, 308, 312, 316, 318, 332, 344, 348, 380, 384, 396, 410, 412, 416, 420, 424, 426, 434, 438, 450, 460, 462, 468, 500, 502, 508, 518, 520, 524, 528, 532, 534, 544, 550, 552, 572, 574, 578, 582, 586, 588, 596
Offset: 1

Views

Author

Harvey P. Dale

Keywords

Links

Crossrefs

Cf. A031443, A031944.
Cf. A049354-A049360. See also A061854, A037861.
Cf. A062756, A077267, A081603.
Row n = 3 of A378000.

Programs

  • Haskell
    a049354 n = a049354_list !! (n-1)
a049354_list = filter f [1..] where
   f n = t0 == a062756 n && t0 == a081603 n where t0 = a077267 n
-- Reinhard Zumkeller, Aug 09 2014
  • Mathematica
    Select[Range[600],Length[Union[DigitCount[#,3]]]== 1&]
FromDigits[#,3]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},3n,{1,0,2}],{n,3}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)
Select[Range@5000, Differences@DigitCount[#,3]=={0,0}&] (* Hans Rudolf Widmer, Dec 11 2021 *)
  • Python
    from sympy.ntheory import count_digits
def ok(n): c = count_digits(n, 3); return c[0] == c[1] == c[2]
print([k for k in range(600) if ok(k)]) # Michael S. Branicky, Nov 15 2021

Formula

A062756(a(n)) = A077267(a(n)) and A081603(a(n)) = A077267(a(n)). - Reinhard Zumkeller, Aug 09 2014

A036990 Numbers n such that, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 64, 66, 68, 72, 74, 76, 80, 82, 84, 88, 96, 98, 100, 104, 112, 128, 130, 132, 136, 138, 140, 144, 146, 148, 152, 160, 162, 164, 168, 170, 172, 176, 178, 180, 184, 192, 194, 196, 200, 202, 204
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

A036989(a(n)) = 1. - Reinhard Zumkeller, Jul 31 2013

Links

Crossrefs

Cf. A036988, A036991, A036992, A061854, A125086.
Each term is 2^n * some term of A014486 (n >= 0).
Cf. A030308.

Programs

  • Haskell
    a036990 n = a036990_list !! (n-1)
a036990_list = filter ((== 1) . a036989) [0..]
-- Reinhard Zumkeller, Jul 31 2013
  • Mathematica
    fQ[n_] := Block[{od = ev = k = 0, id = Reverse@IntegerDigits[n, 2], lmt = Floor@Log[2, n] + 1}, While[k < lmt && od < ev + 1, If[OddQ@id[[k + 1]], od++, ev++ ]; k++ ]; If[k == lmt && od < ev + 1, True, False]]; Select[ Range[0, 204, 2], fQ@# &] (* Robert G. Wilson v, Jan 11 2007 *)
(* b = A036989 *) b[0] = 1; b[n_?EvenQ] := b[n] = Max[b[n/2]-1, 1]; b[n_] := b[n] = b[(n-1)/2]+1; Select[Range[0, 300, 2], b[#] == 1 &] (* Jean-François Alcover, Nov 05 2013, after Reinhard Zumkeller *)

Formula

a(n) = 2*A095775(n). - Robert G. Wilson v

Extensions

More terms from Erich Friedman.

A037016 Numbers n with property that reading binary expansion from right to left (least significant to most significant), run lengths do not decrease.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 12, 13, 14, 15, 21, 25, 26, 28, 29, 30, 31, 42, 50, 51, 53, 56, 57, 58, 60, 61, 62, 63, 85, 101, 102, 106, 113, 114, 115, 117, 120, 121, 122, 124, 125, 126, 127, 170, 202, 204, 205, 213, 226, 227, 229, 230, 234, 240, 241, 242, 243, 245, 248
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

There are A000041(k) elements of this list consisting of k bits: a partition of k written in nonincreasing order corresponds to the binary expansion which when read left to right has run lengths as listed in the partition (reading left to right forces the initial run to be of 1s). - Jason Kimberley, Feb 08 2013
This sequence is a subsequence of A061854 (if we allow the initial 0 to be represented by the empty bit string). - Jason Kimberley, Feb 08 2013
The positive entries are those n for which row n of A101211 is weakly decreasing. Example: 6 is in the sequence because row 6 of A101211 is [2,1]; 8 is not in the sequence because row 8 of A101211 is [1,3]. - Emeric Deutsch, Jan 21 2018

Links

Crossrefs

Cf. A037015 (subsequence), A037014, A037013.

Programs

  • Haskell
    import Data.List (unfoldr, group)
a037016 n = a037016_list !! (n-1)
a037016_list = 0 : filter
   (all (>= 0) . (\x -> zipWith (-) (tail $ rls x) $ rls x)) [1..] where
       rls = map length . group . unfoldr
             (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Mar 10 2012
  • Mathematica
    Select[ Range[0, 250], OrderedQ[ Reverse[ Length /@ Split[ IntegerDigits[#, 2] ] ] ]&] (* Jean-François Alcover, Apr 05 2013 *)

Extensions

More terms from Patrick De Geest, Feb 15 1999
Offset fixed by Reinhard Zumkeller, Mar 10 2012

A072601 Numbers which in base 2 have at least as many 1's as 0's.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 23 2002

Keywords

Examples

			8 = 1000_2 is not present (one '1', three '0's).
10 is present because 10=1010_2 contains 2 '0's and 2 '1's: 2<=2;
11 is present because 11=1011_2 contains 1 '0' and 3 '1's: 1<=3.

Links

Crossrefs

Cf. A007088, A000120, A023416, A061854.
Cf. A037861(a(n)) <= 0.
Cf. A072600 (#0's < #1's), this seq (#0's <= #1's), A031443 (#0's = #1's).
Cf. A072602 (#0's >= #1's), A072603 (#0's > #1's), A044951 (#0's <> #1's).

Programs

  • Haskell
    a072601 n = a072601_list !! (n-1)
a072601_list = filter ((<= 0) . a037861) [0..]
-- Reinhard Zumkeller, Aug 01 2013
  • Mathematica
    geQ[n_] := Module[{a, b}, {a, b} = DigitCount[n, 2]; a >= b]; Select[Range[103], geQ] (* T. D. Noe, Apr 20 2013 *)
Select[Range[110],DigitCount[#,2,1]>=DigitCount[#,2,0]&] (* Harvey P. Dale, Aug 12 2023 *)
  • PARI
    is(n)=2*hammingweight(n)>exponent(n) \\ Charles R Greathouse IV, Apr 18 2020

A061855 Symmetric totally balanced binary sequences: those terms of A014486 which are equal to their reversed complement.

Original entry on oeis.org

0, 2, 10, 12, 42, 52, 56, 170, 178, 204, 212, 232, 240, 682, 722, 738, 812, 852, 868, 920, 936, 976, 992, 2730, 2762, 2866, 2898, 2978, 3010, 3244, 3276, 3380, 3412, 3492, 3524, 3640, 3672, 3752, 3784, 3888, 3920, 4000, 4032, 10922, 11082, 11146
Offset: 0

Views

Author

Antti Karttunen, May 11 2001

Keywords

Comments

These encode symmetric (palindromic) structures in many of the Catalan families, e.g. mountain ranges, parenthesizations, unlabeled rooted plane trees.

Examples

			E.g. the 45th term 11146 is 10101110001010 in binary and can be interpreted as a parenthesization: ( )( )((( )))( )( )

Links

Crossrefs

Obtained by "reflecting" the terms of A061854. Cf. also A035928 (ReflectBinSeq), A061856, A069766.

Programs

  • Maple
    map(op,[seq(PalTotBalBinSequences(j),j=1..10)]);
PalTotBalBinSequences := n -> map(ReflectBinSeq,NonDivingLatticeSequences(n), n);

Formula

a(0) = 0 and the rest with the Maple function map(op, [seq(PalTotBalBinSequences(j), j=1..10)]);

A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 5, 2, 2, 3, 2, 2, 11, 1, 2, 6, 2, 3, 10, 2, 2, 3, 5, 2, 10, 3, 2, 13, 2, 1, 10, 2, 11, 6, 2, 2, 10, 3, 2, 13, 2, 2, 45, 2, 2, 3, 5, 5, 10, 2, 2, 13, 10, 3, 10, 2, 2, 14, 2, 2, 43, 1, 10, 13, 2, 2, 10, 11, 2, 7, 2, 2, 44, 2, 11, 12, 2, 3, 21, 2, 2, 14, 10
Offset: 1

Views

Author

Hartmut F. W. Hoft, Mar 29 2022

Keywords

Comments

The phrase "symmetric representation of sigma(n)" is abbreviated below as SRS(n).
Every number in this sequence is a nondiving number and therefore in A061854. Number 22 with binary pattern 10110 is the smallest nondiving number in A061854, but not in this sequence since a number n with 5 odd divisors must have the form n = 2^m * p^4 for some prime p and some m>=0, and the pattern 10110 of odd/even positions of 1's in a row of A237048 requires 1's at positions 1 < 2^(m+1) < p < p^2 < 2^(m+1) * p <= row(n), a contradiction.
a(2^n) = 1 for all n>=0. The single part of SRS(2^n) has width 1, see A238443.
a(2^m * p) = 3 for odd primes p < 2^(m+1) with m >= 1. SRS(2^m * p) consists of a single part whose 2 subparts have sizes 2*T(n, 1) - 1 = 2^m * p - 1 and 2*T(n, p) - 1 = 2^m - p where T(n, k) = ceiling((n+1)/k -(k+1)/2), see A235791. The numbers 2^m * p are a subsequence of A174973 = A238443.
a(p^k) = A000975(k+1) for all odd primes p and k >= 0. Number a(p^k) in binary has k+1 digits with 1's and 0's alternating. SRS(p^k) has k+1 parts all of width 1 and of the symmetric sizes T(p^k, p^i) - T(p^k, 2*p^i) = (p^(k-i) + p^i)/2, for 0 <= i <= k. The numbers p^k are a subsequence of A174905, the odd primes p form the 1st column in the irregular triangle of A239929 and the numbers p^2 form the 1st column in the irregular triangle of A247687.

Examples

			Sequence values for the first 4 powers of 3: {a(1), a(3), a(9), a(27)} = {1, 2, 5, 10} = {1, 10, 101, 1010}.
Table for a(1..16), a(27) and a(28) together with their lists of the base-2 representation, of the odd/even positions of 1's in the n-th row of A237048, and of the sizes of the parts in SRS(n):
n  a(n) odd/even   A237048         A237270
1   1   {1}        {1}             {1}
2   1   {1}        {1}             {3}
3   2   {1,0}      {1,1}           {2,2}
4   1   {1}        {1,0}           {7}
5   2   {1,0}      {1,1}           {3,3}
6   3   {1,1}      {1,0,1}         {12}
7   2   {1,0}      {1,1,0}         {4,4}
8   1   {1}        {1,0,0}         {15}
9   5   {1,0,1}    {1,1,1}         {5,3,5}
10  2   {1,0}      {1,0,0,1}       {9,9}
11  2   {1,0}      {1,1,0,0}       {6,6}
12  3   {1,1}      {1,0,1,0}       {28}
13  2   {1,0}      {1,1,0,0}       {7,7}
14  2   {1,0}      {1,0,0,1}       {12,12}
15 11   {1,0,1,1}  {1,1,1,0,1}     {8,8,8}
16  1   {1}        {1,0,0,0,0}     {31}
...
27 10   {1,0,1,0}  {1,1,1,0,0,1}   {14,6,6,14}
28  3   {1,1}      {1,0,0,0,0,0,1} {56}
...

Crossrefs

Cf. A000975, A061854, A174905, A174973, A235791, A237048, A237270, A237591, A237593, A238443, A239929, A247687, A352030.

Programs

  • Mathematica
    (* function a237048[ ] is defined in A237048 *)
b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]]
a352696[n_] := Map[b237048, Range[n]]
a352696[85]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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