A083093 -id:A083093 - cp's OEIS frontend

A206427 Square array 2^(m-1)*(3^n+1), read by antidiagonals.

1, 2, 2, 5, 4, 4, 14, 10, 8, 8, 41, 28, 20, 16, 16, 122, 82, 56, 40, 32, 32, 365, 244, 164, 112, 80, 64, 64, 1094, 730, 488, 328, 224, 160, 128, 128, 3281, 2188, 1460, 976, 656, 448, 320, 256, 256, 9842, 6562, 4376, 2920, 1952, 1312, 896, 640, 512, 512

Offset: 0

Marcus Jaiclin, Feb 07 2012

Rectangular array giving the number of 1's in any row of Pascal's Triangle (mod 3) whose row number has exactly m 1's and n 2's in its ternary expansion (listed by antidiagonals).
a(m,n) is independent of the number of zeros in the ternary expansion of the row number.
a(m,n) gives a non-recursive formula for A206424.

			Initial 5 X 5 block of entries (upper corner is (m,n)=(0,0), m increases down, n increases across):
1    2    5   14   41
2    4   10   28   82
4    8   20   56  164
8   16   40  112  328
16  32   80  224  656
Pascal's Triangle (mod 3), row numbers in ternary:
1     <=  Row 0, m = 0, n = 0, 2^(-1)(3^0 + 1) = #1's = 1
1 1     <=  Row 1, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2
1 2 1     <=  Row 2, m = 0, n = 1, 2^(-1)(3^1 + 1) = #1's = 2
1 0 0 1     <=  Row 10, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2
1 1 0 1 1     <=  Row 11, m = 2, n = 0, 2^1(3^0 + 1) = #1's = 4
1 2 1 1 2 1     <=  Row 12, m = 1, n = 1, 2^0(3^1 + 1) = #1's = 4
1 0 0 2 0 0 1     <=  Row 20, m = 0, n = 1, 2^(-1)(3^1 + 1) = #1's = 2
1 1 0 2 2 0 1 1     <=  Row 21, m = 1, n = 1, 2^0(3^1 + 1) = #1's = 4
1 2 1 2 1 2 1 2 1     <=  Row 22, m = 0, n = 2, 2^(-1)(3^2 + 1) = #1's = 5
1 0 0 0 0 0 0 0 0 1     <=  Row 100, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2

Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5

Cf. A206428, A206424, A227428, A083093, A077267, A062756, A081603.

Cf. A062296, A006047, A007318.

a(m, n) = 2^(m - 1)(3^n + 1).

A206428 Rectangular array, a(m,n) = 2^(m-1)*(3^n-1), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 13, 8, 4, 0, 40, 26, 16, 8, 0, 121, 80, 52, 32, 16, 0, 364, 242, 160, 104, 64, 32, 0, 1093, 728, 484, 320, 208, 128, 64, 0, 3280, 2186, 1456, 968, 640, 416, 256, 128, 0, 9841, 6560, 4372, 2912, 1936, 1280, 832, 512, 256, 0

Offset: 0

Marcus Jaiclin, Feb 07 2012

Number of 2's in any row of Pascal's triangle (mod 3) whose row number has exactly m 1's and n 2's in its ternary expansion.
a(m,n) is independent of the number of zeros in the ternary expansion of the row number.
a(m,n) gives a non-recursive formula for A227428.

			Initial 5 X 5 block of array (upper left corner is (0,0), row index m, column index n):
0    1    4   13   40
0    2    8   26   80
0    4   16   52  160
0    8   32  104  320
0   16   64  208  640
Pascal's Triangle (mod 3), row numbers in ternary:
1     <= Row 0, m=0, n=0, 2^(-1)(3^0-1) = #2's = 0
1 1     <= Row 1, m=1, n=0, 2^0(3^0-1) = #2's = 0
1 2 1     <= Row 2, m=0, n=1, 2^(-1)(3^1-1) = #2's = 1
1 0 0 1     <= Row 10, m=1, n=0, 2^0(3^0-1) = #2's = 0
1 1 0 1 1     <= Row 11, m=2, n=0, 2^1(3^0-1) = #2's = 0
1 2 1 1 2 1     <= Row 12, m=1, n=1, 2^0(3^1-1) = #2's = 2
1 0 0 2 0 0 1     <= Row 20, m=0, n=1, 2^(-1)(3^1-1) = #2's = 1
1 1 0 2 2 0 1 1     <= Row 21, m=1, n=1, 2^0(3^1-1) = #2's = 2
1 2 1 2 1 2 1 2 1     <= Row 22, m=0, n=2, 2^(-1)(3^2-1) = #2's = 4
1 0 0 0 0 0 0 0 0 1     <= Row 100, m=1, n=0, 2^0(3^0-1) = #2's = 0

Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5

Cf. A206427, A206424, A227428, A083093, A077267, A062756, A081603.

Cf. A062296, A006047, A007318.

A213126 Rows of triangle formed using Pascal's rule, except sums in the n-th row are modulo n: T(n,0) = T(n,n) = 1 and T(n,k) = (T(n-1,k-1) + T(n-1,k)) mod n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 4, 1, 2, 1, 4, 1, 1, 5, 5, 3, 3, 5, 5, 1, 1, 6, 2, 0, 6, 0, 2, 6, 1, 1, 7, 8, 2, 6, 6, 2, 8, 7, 1, 1, 8, 5, 0, 8, 2, 8, 0, 5, 8, 1, 1, 9, 2, 5, 8, 10, 10, 8, 5, 2, 9, 1, 1, 10, 11, 7, 1, 6, 8, 6

Offset: 0

Alex Ratushnyak, Jun 06 2012

			Triangle begins:
  1;
  1,  1;
  1,  0,  1;
  1,  1,  1,  1;
  1,  2,  2,  2,  1;
  1,  3,  4,  4,  3,  1;
  1,  4,  1,  2,  1,  4,  1;
  1,  5,  5,  3,  3,  5,  5,  1;
  1,  6,  2,  0,  6,  0,  2,  6,  1;
  1,  7,  8,  2,  6,  6,  2,  8,  7,  1;
  1,  8,  5,  0,  8,  2,  8,  0,  5,  8,  1;
  1,  9,  2,  5,  8, 10, 10,  8,  5,  2,  9,  1;

Cf. A007318 - Pascal's triangle read by rows.

Cf. A047999, A083093, A034931, A034930, A034932, A008975.

T[n_,k_]:=If[k==0 || k==n, 1, Mod[T[n - 1, k - 1] + T[n- 1, k], n]]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 29 2017 *)

src = [0]*1024
dst = [0]*1024
for n in range(19):
    dst[0] = dst[n] = 1
    for k in range(1, n):
        dst[k] = (src[k-1]+src[k]) % n
    for k in range(n+1):
        src[k] = dst[k]
        print(dst[k], end=',')

Offset corrected by Joerg Arndt, Dec 05 2016

A386441 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 27.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 8, 8, 21, 7, 1, 1, 8, 1, 2, 16, 2, 1, 8, 1, 1, 9, 9, 3, 18, 18, 3, 9, 9, 1, 1, 10, 18, 12, 21, 9, 21, 12, 18, 10, 1, 1, 11, 1, 3, 6, 3, 3, 6, 3, 1, 11, 1, 1, 12, 12, 4, 9, 9, 6, 9, 9, 4, 12, 12, 1, 1, 13, 24, 16, 13, 18, 15, 15, 18, 13, 16, 24, 13, 1

Offset: 0

Chai Wah Wu, Jul 21 2025

			Triangle begins:
               1;
             1,  1;
           1,  2,  1;
         1,  3,  3,  1;
       1,  4,  6,  4,  1;
     1,  5,  10,  10,  5,  1;
   1,  6,  15,  20,  15,  6,  1;
 1,  7,  21,  8,   8,  21,  7,  1;
  ...

Chai Wah Wu, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
Kenneth S. Davis and William A. Webb, Lucas' Theorem for Prime Powers, Europ. J. Combinatorics, Vol. 11, No. 3 (1990), 229-233.

Cf. A007318, A047999, A083093, A034931, A095140, A095141, A095142, A034930, A008975, A095143, A095144, A095145, A034932.

Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930 (m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

T[i_,j_]:=Mod[Binomial[i,j],27]; Table[T[n,k],{n,0,13},{k,0,n}]//Flatten (* Stefano Spezia, Jul 22 2025 *)

from math import isqrt, comb
from sympy import multiplicity
from gmpy2 import digits
def A386441(n):
    def g1(s,w,e):
        c, d = 1, 0
        if len(s) == 0: return c, d
        a, b = int(s,3), int(w,3)
        if a>=b:
            k = comb(a,b)%27
            j = multiplicity(3,k)
            d += j*e
            k = k//3**j
            c = c*pow(k,e,27)%27
        else:
            if int(s[0:1],3)4: return 0
    s = s.zfill(3)
    w = w.zfill(l:=len(s))
    c, d = g1(s[:3],w[:3],1)
    for i in range(1,l-2):
        c0, d0 = g1(s[i:i+3],w[i:i+3],1)
        c1, d1 = g1(s[i:i+2],w[i:i+2],-1)
        c = c*c0*c1%27
        d += d0+d1
    return c*3**d%27

T(i, j) = binomial(i, j) mod 27.

A178206 Decimal representation of asymptotic growth constant for the number of acyclic orientations on the two-dimensional Sierpinski gasket SG2(n) in the large n limit.

Original entry on oeis.org

1, 1, 2, 7, 2, 9, 9, 0, 7, 0, 5, 3, 6, 6, 1, 6

Offset: 1

Jonathan Vos Post, May 22 2010

Proposition III.1, p.10 of Chang. The paper also studies the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket $SG_{2,b}(n)$ at stage $n$ with $b$ equal to two and three, and determines the asymptotic behaviors. It also derives upper bounds for the asymptotic growth constants for $SG_{2,b}$ and $d$-dimensional Sierpinski gasket $SG_d$.

			1.127299070536616....

Shu-Chiuan Chang, Acyclic orientations on the Sierpinski gasket, May 20, 2010.

Cf. A007318, A054431, A001317, A008292, A083093, A034931, A034930, A008975, A034932, A047999.

A206425 Erroneous version of A227428.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 2, 4, 0, 0, 2, 0, 0, 4, 2, 4, 8, 1, 2, 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

Offset: 0

Marcus Jaiclin, Feb 07 2012

dead

A006047(n) = A206424(n) + a(n).

			Example: Rows 0-8 of Pascal's Triangle (mod 3) are:
1                   So a(0) = 0
1 1                 So a(1) = 0
1 2 1               So a(2) = 1
1 0 0 1                 .
1 1 0 1 1               .
1 2 1 1 2 1             .
1 0 0 2 0 0 1
1 1 0 2 2 0 1 1
1 2 1 2 1 2 1 2 1

Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5

Cf. A083093, A062296, A006047.

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

cp's OEIS Frontend

A206427 Square array 2^(m-1)*(3^n+1), read by antidiagonals.

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A206428 Rectangular array, a(m,n) = 2^(m-1)*(3^n-1), read by antidiagonals.

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A213126 Rows of triangle formed using Pascal's rule, except sums in the n-th row are modulo n: T(n,0) = T(n,n) = 1 and T(n,k) = (T(n-1,k-1) + T(n-1,k)) mod n.

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A386441 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 27.

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A178206 Decimal representation of asymptotic growth constant for the number of acyclic orientations on the two-dimensional Sierpinski gasket SG2(n) in the large n limit.

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A206425 Erroneous version of A227428.

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