A083094 -id:A083094 - cp's OEIS frontend

A083095 a(n) = A083094(n)/4.

0, 2, 5, 6, 14, 15, 18, 20, 41, 42, 45, 47, 54, 56, 59, 60, 122, 123, 126, 128, 135, 137, 140, 141, 162, 164, 167, 168, 176, 177, 180, 182, 365, 366, 369, 371, 378, 380, 383, 384, 405, 407, 410, 411, 419, 420, 423, 425, 486, 488, 491, 492, 500

Offset: 1

Benoit Cloitre, Apr 22 2003

Is this the same as A083097? - Andrew S. Plewe, May 30 2007

Hugo Pfoertner, Table of n, a(n) for n = 1..16384 (terms 1..106 from Paul F. Marrero Romero)

Cf. A010060, A051638, A083093, A083094, A083097, A074939.

Select[Range[0,2000], OddQ[Sum[Mod[Binomial[#,j],3],{j,0,#}]]&]/4 (* Paul F. Marrero Romero, Dec 28 2024 *)

def A083095(n): return int(bin(((m:=n-1).bit_count()&1)+(m<<1))[2:],3)>>1 # Chai Wah Wu, Jun 26 2025

Apparently (2*a(n)) mod 3 = A010060(n-1), the Thue-Morse sequence.
Numbers k such that C(4*k, 2*k) == 1 (mod 3). - Benoit Cloitre, Jul 30 2003
Numbers k such that the base-3 digits of 2k contains no 2's, i.e. a(n) = A074939(n-1)/2. - Chai Wah Wu, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003

A074939 Even numbers such that base 3 representation contains no 2.

Original entry on oeis.org

0, 4, 10, 12, 28, 30, 36, 40, 82, 84, 90, 94, 108, 112, 118, 120, 244, 246, 252, 256, 270, 274, 280, 282, 324, 328, 334, 336, 352, 354, 360, 364, 730, 732, 738, 742, 756, 760, 766, 768, 810, 814, 820, 822, 838, 840, 846, 850, 972, 976, 982, 984, 1000, 1002

Offset: 0

Benoit Cloitre, Oct 04 2002; Nov 15 2003

Even numbers in A005836; n such that binomial(2n,n) == 1 (mod 3).

Sum of an even number of distinct powers of 3. - Emeric Deutsch, Dec 03 2003

Amiram Eldar, Table of n, a(n) for n = 0..10000
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Intersection of A005843 and A005836.

Cf. A006996, A010060, A055246, A074938, A083095.

Select[2*Range[0,600],DigitCount[#,3,2]==0&] (* Harvey P. Dale, Dec 10 2016 *)

def A074939(n): return int(bin((n.bit_count()&1)+(n<<1))[2:],3) # Chai Wah Wu, Jun 26 2025

a(n) = A083094(n)/2; a(n) mod 3 = A010060(n); n such that coefficient of x^n equals 1 in Product_{k>=0} (1 - x^(3^k)).

a(n) + A074938(n) = A055246(n+1). - Philippe Deléham, Jul 10 2005

A122485 Values of A083097(k) such that A083097(k) = A083097(k+1) - 1.

Original entry on oeis.org

5, 14, 41, 59, 122, 140, 167, 176, 365, 383, 410, 419, 491, 500, 527, 545, 1094, 1112, 1139, 1148, 1220, 1229, 1256, 1274, 1463, 1472, 1499, 1517, 1580, 1598, 1625, 1634, 3281, 3299, 3326, 3335, 3407, 3416, 3443, 3461, 3650, 3659, 3686, 3704, 3767, 3785, 3812

Offset: 1

Alexander Adamchuk, Sep 15 2006

nonn

A083097(n) = A083095(n) = A083096(n)/6 = A083094(n)/4, where A083096 are the Numbers k such that 3 divides Sum_{j=1..k} C(2*j,j) = A066796(k).
All terms are of the form 9*m + 5 and belong to A017221 with m = {0, 1, 4, 6, 13, 15, 18, 19, 40, 42, ...}.
Corresponding numbers m such that a(m) = A083097(m) are A129771 (evil odd numbers).

			A083097 begins {0, 2, 5, 6, 14, 15, 18, 20, 41, 42, 45, 47, 54, 56, 59, 60, ...}.
So a(1) = 5 because 5 = A083097(3) = A083097(3+1) - 1.
a(2) = 14 because 14 = A083097(5) = A083097(5+1) - 1.

Cf. A000069, A000120, A066796, A083094, A083095, A083096, A083097, A010060, A129771.

a(n) = A083097(A129771(n)).

More terms from R. J. Mathar, Jan 17 2008
More terms from Jinyuan Wang, Jan 22 2022
Edited by Michel Marcus, Jan 22 2022

cp's OEIS Frontend

A083095 a(n) = A083094(n)/4.

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A074939 Even numbers such that base 3 representation contains no 2.

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A122485 Values of A083097(k) such that A083097(k) = A083097(k+1) - 1.

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