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
Examples
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)
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.
Programs
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Magma
/* As triangle */ [[2^(n+1) + 2^k - 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 27 2017
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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 *)
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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
Formula
As triangle, T(n,k) = 2^(n+1) + 2^k - 1 = A099627(n+1, k).
Extensions
Offset changed to 1 by N. J. A. Sloane, Jul 27 2017
Comments