A083095 -id:A083095 - cp's OEIS frontend

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

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

A083097 a(n) = A083096(n)/6.

Original entry on oeis.org

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 A083095? - Andrew S. Plewe, May 30 2007

Apparently (2*a(n)) mod 3 = A010060(n-1), the Thue-Morse sequence.

Michel Marcus, Table of n, a(n) for n = 1..8192

Cf. A010060, A083095, A083096.

Join[{0},Position[Accumulate[Table[Binomial[2 j, j], {j, 3000}]], ?(Divisible[#, 3] &)]//Flatten]/6 (* _Paul F. Marrero Romero, Dec 30 2024 *)

A083094 Numbers k such that Sum_{j=0..k} (binomial(k,j) mod 3) is odd.

Original entry on oeis.org

0, 8, 20, 24, 56, 60, 72, 80, 164, 168, 180, 188, 216, 224, 236, 240, 488, 492, 504, 512, 540, 548, 560, 564, 648, 656, 668, 672, 704, 708, 720, 728, 1460, 1464, 1476, 1484, 1512, 1520, 1532, 1536, 1620, 1628, 1640, 1644, 1676, 1680, 1692, 1700, 1944, 1952

Offset: 1

Benoit Cloitre, Apr 22 2003

Apparently a(n)/2 (mod 3) = A010060(n), the Thue-Morse sequence.

Cf. A010060, A051638, A074939, A083093, A083095 (gives a b-file of 16384 terms).

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

isok(n) = sum(k=0, n, binomial(n,k) % 3) % 2; \\ Michel Marcus, Dec 05 2013

def A083094(n): return int(bin(((m:=n-1).bit_count()&1)+(m<<1))[2:],3)<<1 # Chai Wah Wu, Jun 26 2025

a(n) = 4*A083095(n). - Hugo Pfoertner, Jan 12 2025

Numbers that are multiples of 4 and such that base-3 digits contain no 1's, or equivalently, numbers such that base-3 digits contains an even number of 2's and no 1's, i.e. a(n) = 2*A074939(n-1). This characterization can be derived from the formula in A051638. - Chai Wah Wu, Jun 26 2025

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

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

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

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A083097 a(n) = A083096(n)/6.

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A083094 Numbers k such that Sum_{j=0..k} (binomial(k,j) mod 3) is odd.

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

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