A006047 -id:A006047 - cp's OEIS frontend

A227428 Number of twos in row n of triangle A083093.

0, 0, 1, 0, 0, 2, 1, 2, 4, 0, 0, 2, 0, 0, 4, 2, 4, 8, 1, 2, 4, 2, 4, 8, 4, 8, 13, 0, 0, 2, 0, 0, 4, 2, 4, 8, 0, 0, 4, 0, 0, 8, 4, 8, 16, 2, 4, 8, 4, 8, 16, 8, 16, 26, 1, 2, 4, 2, 4, 8, 4, 8, 13, 2, 4, 8, 4, 8, 16, 8, 16, 26, 4, 8, 13, 8, 16, 26, 13, 26, 40

Offset: 0

Reinhard Zumkeller, Jul 11 2013

nonn

"The number of entries with value r in the n-th row of Pascal's triangle modulo k is found to be 2^{#r^k (n)}, where now #_r^k (n) gives the number of occurrences of the digit r in the base-k representation of the integer n." [Wolfram] - _R. J. Mathar, Jul 26 2017 [This is not correct: there are entries in the sequence that are not powers of 2. - Antti Karttunen, Jul 26 2017]

			Example of Wilson's formula: a(26) = 13 = 2^(0-1)*(3^3-1) = 26/2, where A062756(26)=0, A081603(26)=3, 26=(222)_3. - _R. J. Mathar_, Jul 26 2017

Reinhard Zumkeller (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..19683
R. Garfield and H. S. Wilf, The distribution of the binomial coefficients modulo p, J. Numb. Theory 41 (1) (1992) 1-5.
Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5
D. L. Wells, Residue counts modulo three for the fibonacci triangle, Appl. Fib. Numbers, Proc. 6th Int Conf Fib. Numbers, Pullman, 1994 (1996) 521-536.
Avery Wilson, Pascal's Triangle Modulo 3, Mathematics Spectrum, 47-2 - January 2015, pp. 72-75.
S. Wolfram, Geometry of binomial coefficients, Am. Math. Monthly 91 (9) (1984) 566-571.

Cf. A006047, A062296, A062756, A083093, A081603, A206424, A206428.

a227428 = sum . map (flip div 2) . a083093_row

A227428 := proc(n)
    local a;
    a := 0 ;
    for k from 0 to n do
        if A083093(n,k) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A227428(n),n=0..20) ; # R. J. Mathar, Jul 26 2017

Table[Count[Mod[Binomial[n, Range[0, n]], 3], 2], {n, 0, 99}] (* Alonso del Arte, Feb 07 2012 *)

A227428(n) = sum(k=0,n,2==(binomial(n,k)%3)); \\ (Naive implementation, from the description) Antti Karttunen, Jul 26 2017

from sympy import binomial
def a(n):
    return sum(1 for k in range(n + 1) if binomial(n, k) % 3 == 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 26 2017

from sympy.ntheory import digits
def A227428(n):
    s = digits(n,3)[1:]
    return 3**s.count(2)-1<>1 # Chai Wah Wu, Jul 24 2025

(define (A227428 n) (* (A000079 (- (A062756 n) 1)) (+ -1 (A000244 (A081603 n))))) ;; After Wilson's direct formula, Antti Karttunen, Jul 26 2017

a(n) = A006047(n) - A206424(n) = n + 1 - A062296(n) - A206424(n).
a(n) = 2^(N_1-1)*(3^N_2-1) where N_1 = A062756(n), N_2 = A081603(n). [Wilson, Theorem 2, Wells] - R. J. Mathar, Jul 26 2017
a(n) = A206424(n) * ((3^A081603(n))-1) / ((3^A081603(n))+1). - Antti Karttunen, Jul 27 2017
a(n) = (1/2)*Sum_{k = 0..n} mod(C(n,k)^2 - C(n,k), 3). - Peter Bala, Dec 17 2020

A249733 Number of (not necessarily distinct) multiples of 9 on row n of Pascal's triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 4, 2, 0, 2, 1, 0, 12, 6, 0, 8, 4, 0, 4, 2, 0, 24, 21, 18, 19, 14, 9, 14, 7, 0, 28, 20, 12, 20, 13, 6, 12, 6, 0, 32, 19, 6, 21, 12, 3, 10, 5, 0, 48, 42, 36, 38, 28, 18, 28, 14, 0, 50, 37, 24, 36, 24, 12, 22, 11, 0, 52, 32, 12, 34, 20, 6, 16, 8, 0

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

Number of zeros on row n of A095143 (Pascal's triangle reduced modulo 9).

This should have a formula. See for example A062296, A006047 and A048967.

			Row 9 of Pascal's triangle is {1, 9, 36, 84, 126, 126, 84, 36, 9, 1}. The terms 9, 36, and 126 are the only multiples of nine, and each of them occurs two times on that row, thus a(9) = 2*3 = 6.
Row 10 of Pascal's triangle is {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1}. The terms 45 (= 9*5) and 252 (= 9*28) are the only multiples of nine, and the former occurs twice, while the latter is alone at the center, thus a(10) = 2+1 = 3.

Antti Karttunen, Table of n, a(n) for n = 0..6561
James G. Huard, Blair K. Spearman and Kenneth S. Williams, Pascal's triangle (mod 9), Acta Arithmetica (1997), Volume: 78, Issue: 4, page 331-349.

Cf. A007318, A048277, A048967, A062296, A095143, A249343, A249723, A249731, A249732, A051382, A249719, A249720, A006047.

Total/@Table[If[Mod[Binomial[n,k],9]==0,1,0],{n,0,80},{k,0,n}] (* Harvey P. Dale, Feb 12 2020 *)

A249733(n) = { my(c=0); for(k=0,n\2,if(!(binomial(n,k)%9),c += (if(k<(n/2),2,1)))); return(c); } \\ Unoptimized.
for(n=0, 6561, write("b249733.txt", n, " ", A249733(n)));

import re
from gmpy2 import digits
def A249733(n):
    s = digits(n,3)
    n1 = s.count('1')
    n2 = s.count('2')
    n01 = s.count('10')
    n02 = s.count('20')
    n11 = len(re.findall('(?=11)',s))
    n12 = s.count('21')
    return n+1-(((3*(n01+1)+(n02<<2)+n12<<2)+3*n11)*(3**n2<Chai Wah Wu, Jul 24 2025

For all n >= 0, the following holds:
a(n) <= A048277(n).
a(n) <= A062296(n).
a(2*A249719(n)) > 0 and a((2*A249719(n))-1) > 0.
a(n) is odd if and only if n is one of the terms of A249720.

A290094 Restricted growth sequence transform of A290093.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 3, 5, 6, 2, 7, 5, 4, 8, 9, 5, 10, 11, 3, 5, 12, 5, 9, 13, 6, 11, 14, 2, 7, 5, 7, 15, 10, 5, 10, 11, 4, 15, 9, 8, 16, 17, 9, 18, 19, 5, 10, 13, 10, 18, 20, 11, 21, 22, 3, 5, 12, 5, 9, 13, 12, 13, 23, 5, 10, 13, 9, 17, 24, 13, 20, 25, 6, 11, 23, 11, 19, 25, 14, 22, 26, 2, 7, 5, 7, 15, 10, 5, 10, 11, 7, 27, 10, 15, 28, 18, 10, 29, 21, 5, 10, 13

Offset: 0

Antti Karttunen, Jul 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..59049

For all i, j: a(i) = a(j) <=> A290093(n) = A290093(n), thus this matches to all the same base-3 (ternary) related sequences as A290093: A006047, A053735, A062756, A081603, A117942, A206424, A227428, A290091, A290092, A290079, and many others.

A382725 Number of entries in the n-th row of Pascal's triangle not divisible by 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 5, 10, 9, 12, 11, 14, 14, 16, 5, 10, 13, 20, 13, 18, 20, 24, 11, 22, 20, 28, 22, 28, 28, 32, 5, 10, 13, 20, 17, 26, 28, 40, 13, 26, 26, 36, 28, 40, 40, 48, 11, 22, 28, 44, 28, 40, 44, 56, 22, 44, 40, 56, 44, 56, 56, 64, 5, 10, 13, 20, 17, 26, 28, 40, 17, 34, 34, 52, 36, 56, 56, 80, 13

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 8), European Journal of Combinatorics 19:1 (1998), pp. 45-62.

Cf. A001316, A006047, A194459, A382720-A382724.

def A382725(n):
    n1 = n>>1
    n2 = n1>>1
    np = ~n
    n100 = (n2&(~n1)&np).bit_count()
    n110 = (n2&n1&np).bit_count()
    n10 = (n1&np).bit_count()
    return ((n100+1<<3)+(n110<<1)+n10*(n10+3))<>3 # Chai Wah Wu, Aug 10 2025

A089898 Product of (digits of n each incremented by 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 8, 16, 24, 32, 40, 48, 56, 64

Offset: 0

Marc LeBrun, Nov 13 2003

Sum of products of all subsets of digits of n (with the empty subset contributing 1).
Number of nonnegative values k such that the lunar sum of k and n is n.
First 100 values are 10 X 10 multiplication table, read by rows/columns.

			a(12)=6 since (1+1)*(2+1)=2*3=6 and since (1*2)+(1)+(2)+(1)=2+1+2+1=6 and since the lunar sum of 12 with any of the six values {0,1,2,10,11,12} is 12.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Cf. A007954, A087061, A001316, A006047.

a089898 n = if n < 10 then n + 1 else (d + 1) * a089898 n'
            where (n', d) = divMod n 10
-- Reinhard Zumkeller, Jul 06 2014

seq(convert(map(`+`,convert(n,base,10),1),`*`), n = 0 .. 1000); # Robert Israel, Nov 17 2014

a089898[n_Integer] :=
Prepend[Array[Times @@ (IntegerDigits[#] + 1) &, n], 1]; a089898[77] (* Michael De Vlieger, Dec 22 2014 *)

a(n) = my(d=digits(n)); prod(i=1, #d, d[i]+1); \\ Michel Marcus, Apr 06 2014

a(n) = vecprod(apply(x->x+1, digits(n))); \\ Michel Marcus, Feb 01 2023

a(n) = a(floor(n/10))*(1+(n mod 10)). - Robert Israel, Nov 17 2014

G.f. g(x) satisfies g(x) = (10*x^11 - 11*x^10 + 1)*g(x^10)/(x-1)^2. - Robert Israel, Nov 17 2014

A386952 Number of entries in the n-th row of Pascal's triangle not divisible by 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 8, 12, 9, 12, 15, 14, 16, 18, 7, 14, 21, 14, 19, 24, 21, 24, 27, 4, 8, 12, 12, 18, 24, 20, 28, 36, 9, 18, 27, 20, 28, 36, 31, 38, 45, 14, 28, 42, 28, 38, 48, 42, 48, 54, 7, 14, 21, 20, 31, 42, 33, 48, 63, 14, 28, 42, 31, 44, 57, 48

Offset: 0

Chai Wah Wu, Aug 10 2025

nonn

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 9), Acta Arithmetica, 78 (1997), 331-349.

Cf. A001316, A006047, A194459, A382720-A382725.

import re
from gmpy2 import digits
def A386952(n):
    s = digits(n,3)
    n1 = s.count('1')
    n2 = s.count('2')
    n01 = s.count('10')
    n02 = s.count('20')
    n11 = len(re.findall('(?=11)',s))
    n12 = s.count('21')
    return ((3*((1+n01<<2)+n11)+((n02<<2)+n12<<2))*3**n2<>2

A387050 Number of entries in the n-th row of Pascal's triangle not divisible by 16.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 18, 15, 20, 17, 22, 21, 24, 21, 26, 25, 28, 27, 30, 30, 32, 9, 18, 23, 36, 21, 30, 33, 40, 25, 34, 35, 44, 37, 42, 44, 48, 21, 42, 37, 52, 37, 50, 48, 56, 43, 54, 52, 60, 54, 60, 60, 64, 9, 18, 23, 36, 29

Offset: 0

Chai Wah Wu, Aug 15 2025

nonn

Cf. A001316, A006047, A194459, A382720-A382725, A386952.

def A387050(n):
    n1 = n>>1
    n2 = n1>>1
    n3 = n2>>1
    np = ~n
    n10, n100, n110 = (k1:=n1&np).bit_count(), (k2:=(k1>>1)&np).bit_count(), (k3:=n2&k1).bit_count()
    n1100, n1000, n1010, n1110 = (n3&k2).bit_count(), ((k2>>1)&np).bit_count(), ((k1>>2)&k1).bit_count(), (n3&k3).bit_count()
    return n10*(n10*(n10+3)+6*((n100<<2)+n110)+20)//6+((n1000<<2)+n100+n1010+n1100<<2)+n110+n1110+8<>3

A268128 a(n) = (A000123(n) - A001316)/2.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 8, 9, 17, 21, 28, 33, 45, 53, 66, 75, 100, 117, 140, 161, 193, 221, 258, 291, 344, 389, 446, 499, 573, 639, 722, 797, 913, 1013, 1132, 1249, 1393, 1533, 1698, 1859, 2060, 2253, 2478, 2699, 2965, 3223, 3522, 3813, 4173, 4517, 4910, 5299, 5753

Offset: 0

Tom Edgar, Jan 26 2016

nonn

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005704, A006047, A268127.

b[0] = 1; b[n_] := b[n] = b[Floor[n/2]] + b[n - 1];
c[n_] := Sum[Mod[Binomial[n, k], 2], {k, 0, n}];
a[n_] := (b[n] - c[n])/2;
Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Dec 12 2018 *)

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[floor(i/2)])
    return A[n]
[(b(n)-prod(x+1 for x in n.digits(2)))/2 for n in [0..60]]

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/2)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*2^i is the binary representation of n. Then a(n) = (1/2)*(b(n) - c(n)).

A268444 a(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*4^i is the base-4 representation of n.

Original entry on oeis.org

1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12, 16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27, 36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32, 48, 64, 2, 4, 6, 8, 4, 8, 12, 16, 6, 12, 18, 24

Offset: 0

Tom Edgar, Feb 04 2016

a(n) gives the number of 1's in row n of A243756.

			The base-4 representation of 10 is (2,2) so a(10) = (2+1)*(2+1) = 9.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Cf. A001316, A006047, A074848.

a(n) = my(d=digits(n,4)); prod(k=1, #d, d[k]+1); \\ Michel Marcus, Feb 05 2016

from math import prod
from gmpy2 import digits
def A268444(n):
    s = digits(n,4)
    return prod((int(d)+1)**s.count(d) for d in '123') # Chai Wah Wu, Apr 24 2025

[prod(x+1 for x in n.digits(4)) for n in [0..75]]

(define (A268444 n) (if (zero? n) 1 (let ((d (mod n 4))) (* (+ 1 d) (A268444 (/ (- n d) 4)))))) ;; For R6RS standard. Use modulo instead of mod in older Schemes like MIT/GNU Scheme. - Antti Karttunen, May 28 2017

a(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*4^i.

A382721 Number of entries in the n-th row of Pascal's triangle not divisible by 11.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 8, 16, 24, 32

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024.

Cf. A001316, A006047, A194459, A382720-A382722.

from math import prod
from gmpy2 import digits
def A382721(n): return prod(int(d,11)+1 for d in digits(n,11)) # Chai Wah Wu, Aug 10 2025

cp's OEIS Frontend

A227428 Number of twos in row n of triangle A083093.

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A249733 Number of (not necessarily distinct) multiples of 9 on row n of Pascal's triangle.

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A290094 Restricted growth sequence transform of A290093.

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A382725 Number of entries in the n-th row of Pascal's triangle not divisible by 8.

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A089898 Product of (digits of n each incremented by 1).

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A386952 Number of entries in the n-th row of Pascal's triangle not divisible by 9.

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A387050 Number of entries in the n-th row of Pascal's triangle not divisible by 16.

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A268128 a(n) = (A000123(n) - A001316)/2.

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