A097357 -id:A097357 - cp's OEIS frontend

A036554 Numbers whose binary representation ends in an odd number of zeros.

2, 6, 8, 10, 14, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 66, 70, 72, 74, 78, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 114, 118, 120, 122, 126, 128, 130, 134, 136, 138, 142, 146, 150, 152, 154, 158, 160, 162, 166, 168, 170, 174

Offset: 1

Tom Verhoeff

Fraenkel (2010) called these the "dopey" numbers.
Also n such that A035263(n)=0 or A050292(n) = A050292(n-1).
Indices of even numbers in A033485. - Philippe Deléham, Mar 16 2004
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - Philippe Deléham, Mar 16 2004
Indices of even numbers in A007913, in A001511. - Philippe Deléham, Mar 27 2004
This sequence consists of the increasing values of n such that A097357(n) is even. - Creighton Dement, Aug 14 2004
Numbers with an odd number of 2's in their prime factorization (e.g., 8 = 2*2*2). - Mark Dow, Sep 04 2007
Equals the set of natural numbers not in A003159 or A141290. - Gary W. Adamson, Jun 22 2008
Represents the set of CCW n-th moves in the standard Tower of Hanoi game; and terms in even rows of a [1, 3, 5, 7, 9, ...] * [1, 2, 4, 8, 16, ...] multiplication table. Refer to the example. - Gary W. Adamson, Mar 20 2010
Refer to the comments in A003159 relating to A000041 and A174065. - Gary W. Adamson, Mar 21 2010
If the upper s-Wythoff sequence of s is s, then s=A036554. (See A184117 for the definition of lower and upper s-Wythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A036554 is the limit when the upper s-Wythoff operation is iterated.  For example, starting with s=(1,4,9,16,...) = (n^2), we obtain lower and upper s-Wythoff sequences
  a=(1,3,4,5,6,8,9,10,11,12,14,...) = A184427;
  b=(2,7,12,21,31,44,58,74,...) = A184428.
  Then putting s=a and repeating the operation gives
b'=(2,6,8,10,13,17,20,...), which has the same first four terms as A036554. - Clark Kimberling, Jan 14 2011
Or numbers having infinitary divisor 2, or the same, having factor 2 in Fermi-Dirac representation as a product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013
Thus, numbers not in A300841 or in A302792. Equally, sequence 2*A300841(n) sorted into ascending order. - Antti Karttunen, Apr 23 2018

			From _Gary W. Adamson_, Mar 20 2010: (Start)
Equals terms in even numbered rows in the following multiplication table:
(rows are labeled 1,2,3,... as with the Towers of Hanoi disks)
   1,  3,  5,  7,  9, 11, ...
   2,  6, 10, 14, 18, 22, ...
   4, 12, 20, 28, 36, 44, ...
   8, 24, 40, 56, 72, 88, ...
   ...
As shown, 2, 6, 8, 10, 14, ...; are in even numbered rows, given the row with (1, 3, 5, 7, ...) = row 1.
The term "5" is in an odd row, so the 5th Towers of Hanoi move is CW, moving disc #1 (in the first row).
a(3) = 8 in row 4, indicating that the 8th Tower of Hanoi move is CCW, moving disc #4.
A036554 bisects the positive nonzero natural numbers into those in the A036554 set comprising 1/3 of the total numbers, given sufficiently large n.
This corresponds to 1/3 of the TOH moves being CCW and 2/3 CW. Row 1 of the multiplication table = 1/2 of the natural numbers, row 2 = 1/4, row 3 = 1/8 and so on, or 1 = (1/2 + 1/4 + 1/8 + 1/16 + ...). Taking the odd-indexed terms of this series given offset 1, we obtain 2/3 = 1/2 + 1/8 + 1/32 + ..., while sum of the even-indexed terms is 1/3. (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550 (see d(n) on page 501).
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
A. S. Fraenkel, Home Page
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Eric Sopena, i-Mark: A new subtraction division game, arXiv:1509.04199 [cs.DM], 2015.
M. Stoll, Chabauty without the Mordell-Weil group, arXiv preprint arXiv:1506.04286 [math.NT], 2015.
Index entries for 2-automatic sequences.

Indices of odd numbers in A007814. Subsequence of A036552. Complement of A003159. Also double of A003159.

Cf. A000041, A003157, A003158, A005408, A052330, A072939, A079523, A096268 (characteristic function, when interpreted with offset 1), A141290, A174065, A300841.

a036554 = (+ 1) . a079523  -- Reinhard Zumkeller, Mar 01 2012

[2*m:m in [1..100] | Valuation(m,2) mod 2 eq 0]; // Marius A. Burtea, Aug 29 2019

Select[Range[200],OddQ[IntegerExponent[#,2]]&] (* Harvey P. Dale, Oct 19 2011 *)

is(n)=valuation(n,2)%2 \\ Charles R Greathouse IV, Nov 20 2012

def ok(n):
  c = 0
  while n%2 == 0: n //= 2; c += 1
  return c%2 == 1
print([m for m in range(1, 175) if ok(m)]) # Michael S. Branicky, Feb 06 2021

from itertools import count, islice
def A036554_gen(startvalue=1): return filter(lambda n:(~n & n-1).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
A036554_list = list(islice(A036554_gen(),30)) # Chai Wah Wu, Jul 05 2022

is_A036554 = lambda n: A001511(n)&1==0 # M. F. Hasler, Nov 26 2024

def A036554(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 29 2025

a(n) = A079523(n)+1 = A072939(n)-1.
a(n) = A003156(n) + n = A003157(n) - n = A003158(n) - n + 1. - Philippe Deléham, Apr 10 2004
Values of k such that A091297(k) = 2. - Philippe Deléham, Feb 25 2004
a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012 [In fact, a(n) = 3n + O(log n). - Charles R Greathouse IV, Nov 27 2024]
a(n) = 2*A003159(n). - Clark Kimberling, Sep 30 2014
{a(n)} = A052330({A005408(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Aug 26 2019

Incorrect equation removed from formula by Peter Munn, Dec 04 2020

A026300 Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 9, 1, 5, 14, 25, 30, 21, 1, 6, 20, 44, 69, 76, 51, 1, 7, 27, 70, 133, 189, 196, 127, 1, 8, 35, 104, 230, 392, 518, 512, 323, 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835, 1, 10, 54, 200, 560, 1242, 2235, 3288, 3915, 3610, 2188

Offset: 0

Clark Kimberling

Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers.

Consider a semi-infinite chessboard with squares labeled (n,k), ranks or rows n >= 0, files or columns k >= 0; number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A114929, A111808, A114972.

			Triangle starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  2;
  [3] 1, 3,  5,   4;
  [4] 1, 4,  9,  12,   9;
  [5] 1, 5, 14,  25,  30,  21;
  [6] 1, 6, 20,  44,  69,  76,   51;
  [7] 1, 7, 27,  70, 133, 189,  196,  127;
  [8] 1, 8, 35, 104, 230, 392,  518,  512,  323;
  [9] 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835.

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Henry Bottomley, Illustration of initial terms
M. Buckley, R. Garner, S. Lack, and R. Street, Skew-monoidal categories and the Catalan simplicial set, arXiv preprint arXiv:1307.0265 [math.CT], 2013.
L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.
J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, and N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2016.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
A. Luzón, D. Merlini, M. A. Morón, and R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.
A. Roshan, P. H. Jones and C. D. Greenman, An Exact, Time-Independent Approach to Clone Size Distributions in Normal and Mutated Cells, arXiv preprint arXiv:1311.5769 [q-bio.QM], 2013.
M. János Uray, A family of barely expansive polynomials, Eötvös Loránd University (Budapest, Hungary, 2020).
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, PDF.
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, Combinatorics, Probability and Computing, Volume 24, Special Issue 01,January 2015, pp 354-372.
D. Yaqubi, M. Farrokhi D.G., and H. Gahsemian Zoeram, Lattice paths inside a table. I, arXiv:1612.08697 [math.CO], 2016-2017.

Reflected version is in A064189.
Row sums are in A005773.
T(n,n) are Motzkin numbers A001006.
Other columns of T include A002026, A005322, A005323.
Cf. A099323, A097357, A005043, A059738, A027907, A020474, A059738.

a026300 n k = a026300_tabl !! n !! k
a026300_row n = a026300_tabl !! n
a026300_tabl = iterate (\row -> zipWith (+) ([0,0] ++ row) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Oct 09 2013

A026300 := proc(n,k)
   add(binomial(n,2*i+n-k)*(binomial(2*i+n-k,i) -binomial(2*i+n-k,i-1)), i=0..floor(k/2));
end proc: # R. J. Mathar, Jun 30 2013

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}]; Table[ t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 03 2011 *)
t[, 0] = 1; t[n, 1] := n; t[n_, k_] /; k>n || k<0 = 0; t[n_, n_] := t[n, n] = t[n-1, n-2]+t[n-1, n-1]; t[n_, k_] := t[n, k] = t[n-1, k-2]+t[n-1, k-1]+t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2014 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[1/2 - k/2, -k/2, n - k + 2, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Mar 21 2018 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(i=0, k\2, binomial(n, 2*i+n-k)*(binomial(2*i+n-k, i)-binomial(2*i+n-k, i-1))), ", ");); print(););} \\ Michel Marcus, Jul 25 2015

T(n,k) = Sum_{i=0..floor(k/2)} binomial(n, 2i+n-k)*(binomial(2i+n-k, i) - binomial(2i+n-k, i-1)). - Herbert Kociemba, May 27 2004
T(n,k) = A027907(n,k) - A027907(n,k-2), k<=n.
Sum_{k=0..n} (-1)^k*T(n,k) = A099323(n+1). - Philippe Deléham, Mar 19 2007
Sum_{k=0..n} (T(n,k) mod 2) = A097357(n+1). - Philippe Deléham, Apr 28 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Nov 28 2009
T(n,k) = binomial(n, k)*hypergeom([1/2 - k/2, -k/2], [n - k + 2], 4). - Peter Luschny, Mar 21 2018
T(n,k) = [t^(n-k)] [x^n] 2/(1 - (2*t + 1)*x + sqrt((1 + x)*(1 - 3*x))). - Peter Luschny, Oct 24 2018
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + x + x^2)^n expanded about the point x = 0. - Peter Bala, Feb 26 2023

Corrected and edited by Johannes W. Meijer, Oct 05 2010

cp's OEIS Frontend

A110565 Results from a change in the rules leading to sequence A097357.

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A036554 Numbers whose binary representation ends in an odd number of zeros.

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A026300 Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).

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A044446 Numbers k such that string 1,0 occurs in the base-4 representation of k but not of k+1.

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A128810 Triangle formed by reading triangle A064189 mod 2 .

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