A099628 Numbers m where m-th Catalan number A000108(m) = binomial(2m,m)/(m+1) is divisible by 2 but not by 4, i.e., where A048881(m) = 1.

2, 4, 5, 8, 9, 11, 16, 17, 19, 23, 32, 33, 35, 39, 47, 64, 65, 67, 71, 79, 95, 128, 129, 131, 135, 143, 159, 191, 256, 257, 259, 263, 271, 287, 319, 383, 512, 513, 515, 519, 527, 543, 575, 639, 767, 1024, 1025, 1027, 1031, 1039, 1055, 1087, 1151, 1279, 1535, 2048

Offset

1,1

Comments

Also, there is exactly one digit position in which both a(n)+1 and a(n)-1, written in binary, have a 1; i.e., the bitwise AND of a(n)-1 and a(n)+1 is 2^k, with k > 0. - Wouter Meeussen, Nov 24 2007

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11175 (rows 2..150)

Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.

Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..1830.

Michael De Vlieger, Bitmap showing the binary expansion of a(n) n = 1..300 (24 rows), bits arranged from least to most significant from bottom, n increasing toward the right, where black = 1 and white = 0.

Formula

As triangle, T(n,k) = 2^(n+1) + 2^k - 1 = A099627(n+1, k).

Example

As triangle, rows start
   2;
   4,  5;
   8,  9, 11;
  16, 17, 19, 23;
  32, 33, 35, 39, 47;
  ...
5 is in the sequence since 10!/(5!6!) = 42 is divisible by 2 but not 4;
6 is not in the sequence since 12!/(6!7!) = 132 is divisible by 4;
7 is not in the sequence since 14!/(7!8!) = 429 is not divisible by 2.
From Michael De Vlieger, Dec 28 2022: (Start)
Table showing the binary expansion of a(n) for n = 1..15, replacing 0 with "." to accentuate the pattern of bits:
   n  a(n)  a(n)_2
  ----------------
   1    2       1.
   2    4      1..
   3    5      1.1
   4    8     1...
   5    9     1..1
   6   11     1.11
   7   16    1....
   8   17    1...1
   9   19    1..11
  10   23    1.111
  11   32   1.....
  12   33   1....1
  13   35   1...11
  14   39   1..111
  15   47   1.1111 (End)

Mathematica

Select[Range[2048], IntegerQ[Log[2, BitAnd[ #+1, #-1]]]&] (* Wouter Meeussen, Nov 24 2007 *)
Table[2^(n + 1) + 2^k - 1, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 28 2022 *)
Select[Range[2100], Boole[Divisible[CatalanNumber[#], {2, 4}]]=={1, 0}&] (* Harvey P. Dale, Jan 31 2024 *)

Prog

(Magma) /* As triangle */ [[2^(n+1) + 2^k - 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 27 2017
(Python)
from itertools import count, islice
def A099628_gen(): # generator of terms
    m = 1
    for n in count(1):
        m *= 2
        r, k = m-1, 1
        for _ in range(n):
            yield r+k
            k *= 2
A099628_list = list(islice(A099628_gen(), 40)) # Chai Wah Wu, Nov 15 2022

Crossrefs

Cf. A000108, A048881, A099627.

Sequence in context: A334265 A334266 A211967 * A346111 A286805 A188072

Adjacent sequences: A099625 A099626 A099627 * A099629 A099630 A099631

Keyword

easy,nonn,tabl

Author

Henry Bottomley, Oct 25 2004

Extensions

Offset changed to 1 by N. J. A. Sloane, Jul 27 2017

Status

approved