A083097 -id:A083097 - cp's OEIS frontend

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

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

A083096 Numbers k such that 3 divides Sum_{j=1..k} binomial(2*j,j).

Original entry on oeis.org

0, 12, 30, 36, 84, 90, 108, 120, 246, 252, 270, 282, 324, 336, 354, 360, 732, 738, 756, 768, 810, 822, 840, 846, 972, 984, 1002, 1008, 1056, 1062, 1080, 1092, 2190, 2196, 2214, 2226, 2268, 2280, 2298, 2304, 2430, 2442, 2460, 2466, 2514, 2520, 2538, 2550

Offset: 1

Benoit Cloitre, Apr 22 2003

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

Hugo Pfoertner, Table of n, a(n) for n = 1..8192

Cf. A066796, A083097, A081601.

Reap[ For[n = 0, n <= 3000, n = n + 3, If[ Divisible[ Sum[ Binomial[2 k, k], {k, 1, n}], 3], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 01 2013 *)
Join[{0},Position[Accumulate[Table[Binomial[2k,k],{k,2600}]],?( Divisible[ #,3]&)]//Flatten] (* _Harvey P. Dale, Mar 14 2020 *)

It appears that sequence gives k such that the coefficient of x^k equals 1 in Product_{j>=1} 1-x^(3^j).

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

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 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

A167912 a(n) = (1/(3^n)^2) * Sum_{k=0..(3^n-1)} binomial(2k,k).

Original entry on oeis.org

1, 217, 913083596083, 18744974860247264575032720770000376335095039

Offset: 1

Alexander Adamchuk, Nov 15 2009

nonn

Note that a(n) mod 27 = a(n) mod 9 = a(n) mod 3 = 1.

The Maple program yields the first seven terms; easily adjustable for obtaining more terms. However, a(4) has 44 digits, a(5) has 140 digits, a(6) has 432 digits and a(7) has 1308 digits. - Emeric Deutsch, Nov 22 2009

Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Eric Weisstein's World of Mathematics, Binomial Sums.

Cf. A006134, A083096, A066796, A083097, A081601, A010060, A122485.

a := proc (n) options operator, arrow: (sum(binomial(2*k, k), k = 0 .. 3^n-1))/3^(2*n) end proc: seq(a(n), n = 1 .. 7); # Emeric Deutsch, Nov 22 2009

Table[(1/3^n)^2 * Sum[Binomial[2 k, k], {k, 0, 3^n - 1}], {n, 1, 5}] (* G. C. Greubel, Jul 01 2016 *)

a(4) from Emeric Deutsch, Nov 22 2009

A181990 a(n) = Sum_{0 <= k <= m < p} (binomial(m, k)^(p-1))/p, where p is the n-th prime.

Original entry on oeis.org

3, 399, 12708885, 124515078454872901983423, 39212583445587381894247266262023061, 43487633454143579523135045521112077473364484383507327790688372131, 157851796824901989964381293031623545741924564754192453966085327785455257503133278729

Offset: 2

Alexander Adamchuk, Apr 04 2012

nonn

a(n) is a sum of all elements in the first p rows of Pascal's triangle each raised to the (p-1) power and divided by p, where p is the n-th prime.

For p = 3 and 7 (and their powers like 3, 9, 27, ... and 7, 49, ...) the sums of all elements in n = p^k top rows of Pascal's triangle each raised to the (n-1) = (p^k-1) power are divisible by n^2 = p^(2k) for all k > 0.

Eric Weisstein's World of Mathematics, Pascal's Triangle
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A007318, A006134, A083096, A066796, A083097, A081601, A010060, A122485, A167912.

Table[(Sum[Binomial[m, k]^(Prime[n] - 1), {m, 0, Prime[n] - 1}, {k, 0, m}])/Prime[n], {n, 2, 10}]

a(n) = my(p=prime(n)); sum(m=0, p-1, sum(k=0, m, binomial(m,k)^(p-1))/p); \\ Michel Marcus, Dec 03 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A083096 Numbers k such that 3 divides Sum_{j=1..k} binomial(2*j,j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A167912 a(n) = (1/(3^n)^2) * Sum_{k=0..(3^n-1)} binomial(2k,k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A181990 a(n) = Sum_{0 <= k <= m < p} (binomial(m, k)^(p-1))/p, where p is the n-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI