A036991 -id:A036991 - cp's OEIS frontend

A036992 Numbers not in A036990 nor A036991.

6, 9, 14, 17, 22, 25, 26, 28, 30, 33, 35, 37, 38, 41, 46, 49, 54, 57, 58, 60, 62, 65, 67, 69, 70, 73, 78, 81, 86, 89, 90, 92, 94, 97, 99, 101, 102, 105, 106, 108, 110, 113, 114, 116, 118, 120, 121, 122, 124, 126, 129, 131, 133, 134, 135, 137, 139, 141, 142, 145, 147

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A036988, A036990, A036991.

a036992 n = a036992_list !! (n-1)
a036992_list = c a036990_list (tail a036991_list) [0..] where
   c us'@(u:us) vs'@(v:vs) (w:ws)
     | w == u    = c us vs' ws
     | w == v    = c us' vs ws
     | otherwise = w : c us' vs' ws
-- Reinhard Zumkeller, Jul 31 2013

a(n) ~ n. [Charles R Greathouse IV, Sep 21 2011]

More terms from Erich Friedman.

A367626 First differences of A036991.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 8, 4, 2, 2, 4, 2, 2, 4, 2, 2, 8, 4, 2, 2, 4, 2, 2, 4, 2, 2, 8, 4, 2, 2, 4, 2, 2, 4, 2, 2, 16, 8, 4, 2, 2, 8, 4, 2, 2, 4, 2, 2, 4, 2, 2, 8, 4, 2, 2, 4, 2, 2, 4, 2, 2, 8, 4, 2, 2, 4, 2, 2, 4, 2, 2, 16, 8, 4, 2, 2, 8, 4, 2, 2

Offset: 1

N. J. A. Sloane, Nov 25 2023

nonn

Entries are powers of 2 (see A036991 and A367627).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A036991, A367625, A367627.

from itertools import count, islice
def A367626_gen(): # generator of terms
    a = 0
    yield 1
    for n in count(1):
        s = bin(n)[2:]
        c, l = 2, len(s)
        for i in range(1,l+1):
            if (c:=c+(2 if s[l-i]=='1' else 0)) <= i:
                break
        else:
            yield n-a<<1
            a = n
A367626_list = list(islice(A367626_gen(),30)) # Chai Wah Wu, Nov 28 2023

A367625 (A036991(n)-1)/2.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 115, 117, 118, 119, 121, 122, 123, 125, 126, 127

Offset: 2

N. J. A. Sloane, Nov 27 2023

nonn

N. J. A. Sloane, Table of n, a(n) for n = 2..13496

Cf. A036771, A367626, A367627.

from itertools import count, islice
def A367625_gen(startvalue=0): # generator of terms >= startvalue
    if startvalue <= 0:
        yield 0
    for n in count(max(startvalue,1)):
        s = bin(n)[2:]
        c, l = 2, len(s)
        for i in range(1,l+1):
            c += int(s[l-i])<<1
            if c <= i:
                break
        else:
            yield n
A367625_list = list(islice(A367625_gen(),30)) # Chai Wah Wu, Nov 28 2023

A350577 Prime numbers in A036991.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 61, 71, 79, 83, 103, 107, 109, 127, 151, 157, 167, 173, 179, 181, 191, 199, 211, 223, 239, 251, 271, 283, 307, 311, 317, 331, 347, 349, 359, 367, 373, 379, 383, 431, 439, 443, 461, 463, 467, 479, 487, 491, 499

Offset: 1

Gennady Eremin, Jan 07 2022

This sequence includes A000668.

Conjecture: The sequence is infinite. For example, in the first million primes (see A000040) 304208 numbers are terms of A036991.

Gennady Eremin, Table of n, a(n) for n = 1..20000
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 10, 14.

Subsequence of A000040, A005408, A036991, A042987, A045395, A095070, A095074, A216285.

Cf. A000668.

q:= proc(n) local l, t, i; l:= Bits[Split](n); t:=0;
      for i to nops(l) do t:= t-1+2*l[i];
        if t<0 then return false fi
      od: true
    end:
select(isprime and q, [$2..500])[];  # Alois P. Heinz, Jan 07 2022

q[n_] := PrimeQ[n] && AllTrue[Accumulate[(-1)^Reverse[IntegerDigits[n, 2]]], # <= 0 &]; Select[Range[500], q] (* Amiram Eldar, Jan 07 2022 *)

from sympy import isprime
def ok(n):
    if n == 0: return True
    count = {"0": 0, "1": 0}
    for bit in bin(n)[:1:-1]:
        count[bit] += 1
        if count["0"] > count["1"]: return False
    return isprime(n)
print([k for k in range(3, 500, 2) if ok(k)]) # Michael S. Branicky, Jan 07 2022

Intersection of A000040 and A036991.

A002054 Binomial coefficient C(2n+1, n-1).

Original entry on oeis.org

1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, 293930, 1144066, 4457400, 17383860, 67863915, 265182525, 1037158320, 4059928950, 15905368710, 62359143990, 244662670200, 960566918220, 3773655750150, 14833897694226, 58343356817424, 229591913401900

Offset: 1

N. J. A. Sloane

a(n) = number of permutations in S_{n+2} containing exactly one 312 pattern. E.g., S_3 has a_1 = 1 permutations containing exactly one 312 pattern, and S_4 has a_2 = 5 permutations containing exactly one 312 pattern, namely 1423, 2413, 3124, 3142, and 4231. This comment is also true if 312 is replaced by any of 132, 213, or 231 (but not 123 or 321, for which see A003517). [Comment revised by N. J. A. Sloane, Nov 26 2022]
Number of valleys in all Dyck paths of semilength n+1. Example: a(2)=5 because UD*UD*UD, UD*UUDD, UUDD*UD, UUD*UDD, UUUDDD, where U=(1,1), D=(1,-1) and the valleys are shown by *. - Emeric Deutsch, Dec 05 2003
Number of UU's (double rises) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDU*UDD, U*UDDUD, U*UDUDD, U*U*UDDD, the double rises being shown by *. - Emeric Deutsch, Dec 05 2003
Number of peaks at level higher than one (high peaks) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUU*DDD, the high peaks being shown by *. - Emeric Deutsch, Dec 05 2003
Number of diagonal dissections of a convex (n+3)-gon into n regions. Number of standard tableaux of shape (n,n,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-1 of which are triangular. Example: a(2)=5 because the convex pentagon ABCDE is dissected by any of the diagonals AC, BD, CE, DA, EB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004
Number of jumps in all full binary trees with n+1 internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch, Jan 18 2007
a(n) is the total number of nonempty Dyck subpaths in all Dyck paths (A000108) of semilength n. For example, the Dyck path UUDUUDDD has Dyck subpaths stretching over positions 1-8 (the entire path), 2-3, 2-7, 4-7, 5-6 and so contributes 5 to a(4). - David Callan, Jul 25 2008
a(n+1) is the total number of ascents in the set of all n-permutations avoiding the pattern 132. For example, a(2) = 5 because there are 5 ascents in the set 123, 213, 231, 312, 321. - Cheyne Homberger, Oct 25 2013
Number of increasing tableaux of shape (n+1,n+1) with largest entry 2n+1. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 5 counts the five tableaux (124)(235), (123)(245), (124)(345), (134)(245), (123)(245). - Oliver Pechenik, May 02 2014
a(n) is the number of noncrossing partitions of 2n+1 into n-1 blocks of size 2 and 1 block of size 3. - Oliver Pechenik, May 02 2014
Number of paths in the half-plane x>=0, from (0,0) to (2n+1,3), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 5 paths: UUUUD, UUUDU, UUDUU, UDUUU, DUUUU. - José Luis Ramírez Ramírez, Apr 19 2015
From Gus Wiseman, Aug 20 2021: (Start)
Also the number of binary numbers with 2n+2 digits and with two more 0's than 1's. For example, the a(2) = 5 binary numbers are: 100001, 100010, 100100, 101000, 110000, with decimal values 33, 34, 36, 40, 48. Allowing first digit 0 gives A001791, ranked by A345910/A345912.
Also the number of integer compositions of 2n+2 with alternating sum -2, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 21 compositions are:
  (35)  (152)  (1124)  (11141)  (111113)
        (251)  (1223)  (12131)  (111212)
               (1322)  (13121)  (111311)
               (1421)  (14111)  (121112)
               (2114)           (121211)
               (2213)           (131111)
               (2312)
               (2411)
The following pertain to these compositions:
- The unordered version is A344741.
- Ranked by A345924 (reverse: A345923).
- A345197 counts compositions by length and alternating sum.
- A345925 ranks compositions with alternating sum 2 (reverse: A345922).
(End)

			G.f. = x + 5*x^2 + 21*x^3 + 84*x^4 + 330*x^5 + 1287*x^6 + 5005*x^7 + ...

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.
George Grätzer, General Lattice Theory. Birkhauser, Basel, 1998, 2nd edition, p. 474, line -3.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
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).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..100 computed by T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2550-2558; preprint, 2017.
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
Arthur Cayley, On the partitions of a polygon, Proc. London Math. Soc., Vol. 22 (1891), pp. 237-262 = Collected Mathematical Papers, Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
Emeric Deutsch, Dyck path enumeration, Discrete Math., Vol. 204, No. 1-3 (1999), pp. 167-202.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318 [math.CO], 2023. See (5) p. 7.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, J. Int. Seq., Vol. 17 (2014), Article 14.1.5; arXiv preprint, arXiv:1203.6792 [math.CO], 2012.
Xiaoyu He, Emily Huang, Ihyun Nam and Rishubh Thaper, Shuffle Squares and Reverse Shuffle Squares, arXiv:2109.12455 [math.CO], 2021.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions.
Werner Krandick, Trees and jumps and real roots, J. Computational and Applied Math., Vol. 162, No. 1 (2004), pp. 51-55.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
Toufik Mansour and Alek Vainshtein, Counting occurrences of 123 in a permutation, arXiv:math/0105073 [math.CO], 2001.
Henri Mühle, Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices, arXiv:1509.06942 [math.CO], 2015.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3.
John Noonan and Doron Zeilberger, The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns, arXiv:math/9808080 [math.CO], 1998.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, Vol. 125 (2014), pp. 357-378.
Karol Penson, Hausdorff moment problems for combinatorial numbers: heuristics via Meijer G-functions, Nov. 2022.
Ronald C. Read, On general dissections of a polygon, Aequationes Mathematicae, Vol. 18, No. 1-2 (1978), pp. 370-388; Preprint, 1974.
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, Vol. 76 , No. 1 (1996), pp. 175-177.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, arXiv:1108.2993 [hep-ph], 2011.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Diagonal 4 of triangle A100257. Also a diagonal of A033282.
Equals (1/2) A024483(n+2). Bisection of A037951 and A037955.
Cf. A001263.
Column k=1 of A263771.
Counts terms of A031445 with 2n+2 digits in binary.
Cf. A000097, A000346, A000984, A001622, A001700, A007318, A008549, A031444, A058622, A097805, A116406, A138364, A163493, A202736.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([1..25],n->Binomial(2*n+1,n-1)); # Muniru A Asiru, Aug 09 2018

[Binomial(2*n+1, n-1): n in [1..30]]; // Vincenzo Librandi, Apr 20 2015

with(combstruct): seq((count(Composition(2*n+2), size=n)), n=1..24); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[8/(((Sqrt[1-4x] +1)^3)*Sqrt[1-4x]), {x,0,22}], x] (* Robert G. Wilson v, Aug 08 2011 *)
a[ n_]:= Binomial[2 n + 1, n - 1]; (* Michael Somos, Apr 25 2014 *)

PARI
```
{a(n) = binomial( 2*n+1, n-1)};
```

from _future_ import division
A002054_list, b = [], 1
for n in range(1,10**3):
    A002054_list.append(b)
    b = b*(2*n+2)*(2*n+3)//(n*(n+3)) # Chai Wah Wu, Jan 26 2016

[binomial(2*n+1, n-1) for n in (1..25)] # G. C. Greubel, Mar 22 2019

a(n) = Sum_{j=0..n-1} binomial(2*j, j) * binomial(2*n - 2*j, n-j-1)/(j+1). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
G.f.: z*C^4/(2-C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch, Jul 05 2003
From Wolfdieter Lang, Jan 09 2004: (Start)
a(n) = binomial(2*n+1, n-1) = n*C(n+1)/2, C(n)=A000108(n) (Catalan).
G.f.: (1 - 2*x - (1-3*x)*c(x))/(x*(1-4*x)) with g.f. c(x) of A000108. (End)
G.f.: z*C(z)^3/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
G.f.: 2F1(5/2, 2; 4; 4*x). - R. J. Mathar, Aug 09 2015
D-finite with recurrence: a(n+1) = a(n)*(2*n+3)*(2*n+2)/(n*(n+3)). - Chai Wah Wu, Jan 26 2016
From Ilya Gutkovskiy, Aug 30 2016: (Start)
E.g.f.: (BesselI(0,2*x) + (1 - 1/x)*BesselI(1,2*x))*exp(2*x).
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). (End)
a(n) = (1/(n+1))*Sum_{i=0..n-1} (n+1-i)*binomial(2n+2,i), n >= 1. - Taras Goy, Aug 09 2018
G.f.: (x - 1 + (1 - 3*x)/sqrt(1 - 4*x))/(2*x^2). - Michael Somos, Jul 28 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=1} 1/a(n) = 5/3 - 2*Pi/(9*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 52*log(phi)/(5*sqrt(5)) - 7/5, where phi is the golden ratio (A001622). (End)
a(n) = A001405(2*n+1) - A000108(n+1), n >= 1 (from Eremin link, page 7). - Gennady Eremin, Sep 05 2023
G.f.: x/(1 - 4*x)^2 * c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Feb 03 2024
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x - 3)/sqrt(4 - x) (see Penson).
G.f. x*/sqrt(1 - 4*x) * c(x)^3. (End)

A052940 a(0) = 1; a(n) = 3*2^n - 1, for n > 0.

Original entry on oeis.org

1, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple regular expression.
Numbers k > 1 such that a(k-1)^2 + a(k) is square, e.g., 5^2 + 11 = 6^2; 11^2 + 23 = 12^2. - Vincenzo Librandi, Aug 06 2010
Numerator of the sum of terms at the n-th level of the Calkin-Wilf tree. - Carl Najafi, Jul 10 2011

Michael De Vlieger, Table of n, a(n) for n = 0..3320
Gennady Eremin, Dyck Numbers, III. Enumeration and bijection with symmetric Dyck paths, arXiv:2302.02765 [math.CO], 2023.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 931
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from initial terms, same as A055010 and A083329.
Subsequence of A036991.
Cf. A000225, A023548, A107909, A134060, A140182.

Concatenation([1], List([1..30], n-> 3*2^n -1)); # G. C. Greubel, Oct 18 2019

[1] cat [3*2^n - 1: n in [1..30]]; // Vincenzo Librandi, Dec 01 2015

spec:= [S,{S=Prod(Sequence(Union(Z,Z)),Union(Sequence(Z),Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0,1,3*2^n -1), n=0..30); # G. C. Greubel, Oct 18 2019

Join[{1},Table[3*2^n-1,{n,30}]] (* or *) Join[{1},LinearRecurrence[{3,-2},{5,11},30]] (* Harvey P. Dale, Mar 07 2015 *)

a(n)=if(n,3*2^n-1,1) \\ Charles R Greathouse IV, Oct 07 2015

Vec((1+2*x-2*x^2)/(-1+2*x)/(-1+x) + O(x^30)) \\ Altug Alkan, Dec 01 2015

print([1] + [(3<Gennady Eremin, Aug 29 2023

[1]+[3*2^n -1 for n in (1..30)] # G. C. Greubel, Oct 18 2019

G.f.: (1+2*x-2*x^2)/((1-x)*(1-2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
Binomial transform of 3 - 0^n - (-1)^n = (1, 4, 2, 4, 2, 4, 2, ...). - Paul Barry, Jun 30 2003
a(n) = A107909(A023548(n+1)) for n > 1. - Reinhard Zumkeller, May 28 2005
Row sums of triangle A134060. - Gary W. Adamson, Oct 05 2007
Equals row sums of triangle A140182. - Gary W. Adamson, May 11 2008
Equals M*Q, where M is a modified Pascal triangle (1,2) with first term "1" instead of 2; as an infinite lower triangular matrix. Q is the vector (1, 2, 2, 2, ...). - Gary W. Adamson, Nov 30 2015
From Gennady Eremin, Aug 29 2023: (Start)
a(n+1) = 2*a(n) + 1 for n > 0.
a(n) = (A000225(n+1) + A000225(n+2))/2 for n > 0. (End)

More terms from James Sellers, Jun 08 2000

a(30)-a(32) from Vincenzo Librandi, Dec 01 2015

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

N. J. A. Sloane

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.

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

a036990 n = a036990_list !! (n-1)
a036990_list = filter ((== 1) . a036989) [0..]
-- Reinhard Zumkeller, Jul 31 2013

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 *)

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

More terms from Erich Friedman.

A061854 Nondiving binary sequences: numbers which in base 2 have at least the same number of 1's as 0's and reading the binary expansion from left (msb) to right (least significant bit), the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Antti Karttunen, May 11 2001

"msb" = "most significant bit", A053644.
These encode lattice walks using steps (+1,+1) (= 1's in binary expansion) and (+1,-1) (= 0's in binary expansion) that start from the origin (0,0) and never "dive" under the "sea-level" y=0.
The number of such walks of length n (here: the terms of binary width n) is given by C(n,floor(n/2)) = A001405, which is based on the fact mentioned in Guy's article that the shallow diagonals of the Catalan triangle A009766 sum to A001405.
From Jason Kimberley, Feb 08 2013: (Start)
This sequence is a subsequence of A072601.
Define a map from this set onto the nonnegative integers as follows: set the output bit string to be empty, representing zero; process the input string from left to right; when 1 occurs, change the rightmost 0 in the output to 1; if there is no 0 in the output, prepend a 1; when 0 occurs in the input, change the rightmost 1 in the output to 1. The definition of this sequence ensures that we always have a 1 in the output when a 0 occurs in the input. We this map is onto by showing the restriction to the subset Asubsequence is onto. (End)
The binary representation of a(n) is the numeric representation of the left half of a symmetric balanced string of parentheses with "(" representing 1 and ")" representing 0 (see comments and examples in A001405). Some of the numbers in this sequence cannot be realized as the 1-0-pattern of the odd/even positions of 1's in any row n of A237048 that determines the parts and their widths in the symmetric representation of sigma(n), see A352696. - Hartmut F. W. Hoft, Mar 29 2022

			From _Hartmut F. W. Hoft_, Mar 29 2022: (Start)
The columns in the table are the numbers n, the base-2 representation of n, the left half of the symmetric balanced string of parentheses corresponding to n, validity of the nondiving property for n, and associated number a(n):
1   1      (      True    a(1)
2   10     ()     True    a(2)
3   11     ((     True    a(3)
4   100    ())    False    -
5   101    ()(    True    a(4)
6   110    (()    True    a(5)
7   111    (((    True    a(6)
8   1000   ()))   False    -
9   1001   ())(   False    -
10  1010   ()()   True    a(7)
...
20  10100  ()())  False    -
21  10101  ()()(  True    a(13)
...
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Antti Karttunen, Some notes on Catalan's Triangle
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A001405, A031443, A036990, A036991, A061855, A014486.

Cf. A237593, A237048, A237591, A237593, A352696.

# We use a simple backtracking algorithm: map(op,[seq(NonDivingLatticeSequences(j),j=1..10)]);
NDLS_GLOBAL := []; NonDivingLatticeSequences := proc(n) global NDLS_GLOBAL; NDLS_GLOBAL := []; NonDivingLatticeSequencesAux(0,0,n); RETURN(NDLS_GLOBAL); end;
NonDivingLatticeSequencesAux := proc(x,h,i) global NDLS_GLOBAL; if(0 = i) then NDLS_GLOBAL := [op(NDLS_GLOBAL),x]; else if(h > 0) then NonDivingLatticeSequencesAux((2*x),h-1,i-1); fi; NonDivingLatticeSequencesAux((2*x)+1,h+1,i-1); fi; end;

a061854[n_] := Select[Range[n], !MemberQ[FoldList[#1+If[#2>0, 1, -1]&, 0, IntegerDigits[n, 2]], -1]]
a061854[116] (* Hartmut F. W. Hoft, Mar 29 2022 *)
Select[Range[120],Min[Accumulate[IntegerDigits[#,2]/.(0->-1)]]>=0&] (* Harvey P. Dale, Sep 11 2023 *)

A367627 a(n) = log_2(A367626(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 25 2023

nonn

First differences of A036991 are powers of 2 (see A036991 and A367626).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A036991, A367626.

from itertools import count, islice
def A367627_gen(): # generator of terms
    a = 0
    yield 0
    for n in count(1):
        s = bin(n)[2:]
        c, l = 2, len(s)
        for i in range(1,l+1):
            if (c:=c+(2 if s[l-i]=='1' else 0)) <= i:
                break
        else:
            yield (n-a).bit_length()
            a = n
A367627_list = list(islice(A367627_gen(),30)) # Chai Wah Wu, Nov 28 2023

A036993 Numbers n with property that reading from right to left in the binary expansion of n, the number of 0's always stays ahead of the number of 1's.

Original entry on oeis.org

0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 64, 68, 72, 80, 84, 88, 96, 100, 104, 112, 128, 132, 136, 144, 148, 152, 160, 164, 168, 176, 192, 196, 200, 208, 224, 256, 260, 264, 272, 276, 280, 288, 292, 296, 304, 320, 324, 328, 336, 340, 344, 352, 356, 360, 368, 384

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Cf. A036990, A036991, A036992, A036994.

Cf. A030308.

a036993 n = a036993_list !! (n-1)
a036993_list = filter ((p 0) . a030308_row) [0..] where
   p _     []     = True
   p zeros (0:bs) = p (zeros + 1) bs
   p zeros (1:bs) = zeros > 1 && p (zeros - 1) bs
-- Reinhard Zumkeller, Jul 31 2013

Select[Range[0,400],Min[Accumulate[Reverse[IntegerDigits[#,2]]/.{0->1, 1->-1}]]>0&] (* Harvey P. Dale, Aug 25 2013 *)

More terms from Erich Friedman

cp's OEIS Frontend

A036992 Numbers not in A036990 nor A036991.

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A367626 First differences of A036991.

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A367625 (A036991(n)-1)/2.

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A350577 Prime numbers in A036991.

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A002054 Binomial coefficient C(2n+1, n-1).

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A052940 a(0) = 1; a(n) = 3*2^n - 1, for n > 0.

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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.

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