A208280 -id:A208280 - cp's OEIS frontend

A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.

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

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A047999, A083093, A034931, A034930, A034932.
Cf. A208278 (row sums), A208279 (central terms), A208134 (number of zeros per row), A208280 (distinct terms per row).
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), (this sequence) (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

a008975 n k = a008975_tabl !! n !! k
a008975_row n = a008975_tabl !! n
a008975_tabl = iterate
   (\row -> map (`mod` 10) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Feb 24 2012

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 10] (* Robert G. Wilson v, May 26 2004 *)

from math import isqrt, comb
from sympy.ntheory.modular import crt
def A008975(n):
    w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1
    d = int(not ~r & w)
    while True:
        r, a = divmod(r,5)
        w, b = divmod(w,5)
        c = c*comb(a,b)%5
        if r<5 and w<5:
            c = c*comb(r,w)%5
            break
    return crt([5,2],[c,d])[0] # Chai Wah Wu, May 01 2025

T(i, j) = binomial(i, j) mod 10.

A368077 Numbers k such that row k of Pascal's triangle mod 10 contains all the numbers 0 to 9.

Original entry on oeis.org

47, 59, 89, 94, 117, 118, 119, 123, 147, 173, 189, 198, 214, 219, 221, 222, 223, 233, 237, 238, 239, 243, 244, 247, 248, 297, 298, 309, 313, 317, 318, 319, 323, 339, 344, 345, 346, 347, 348, 363, 366, 367, 368, 369, 373, 397, 398, 409, 413, 414, 417, 418, 421, 422, 423, 429, 433, 437, 438, 439

Offset: 1

Robert Israel, Dec 10 2023

Numbers k such that A208280(k) = 10.

			a(3) = 89 is a term because
  binomial(89,15) = 38163061637050680 == 0 (mod 10),
  binomial(89,0) = 1 == 1 (mod 10),
  binomial(89,5) = 41507642 == 2 (mod 10),
  binomial(89,8) = 70625252863 == 3 (mod 10),
  binomial(89,3) = 113564 == 4 (mod 10),
  binomial(89,16) = 176504160071359395 == 5 (mod 10),
  binomial(89,2) = 3916 == 6 (mod 10),
  binomial(89,9) = 635627275767 == 7 (mod 10),
  binomial(89,6) = 581106988 == 8 (mod 10), and
  binomial(89,1) = 89 == 9 (mod 10).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A208280.

filter:= proc(n) local k,S;
  S:= {$0..9}:
  for k from 0 to n/2 do
    S:= S minus {(binomial(n,k) mod 10)};
    if S = {} then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 10 2023

from itertools import count, islice
def A368077_gen(): # generator of terms
    a, b = [], set(range(10))
    for i in count(0):
        c, d = 0, []
        for k in a:
            d.append((c+k)%10)
            c = k
        a = d+[1]
        if b.issubset(set(a)): yield i
A368077_list = list(islice(A368077_gen(),30)) # Chai Wah Wu, May 01 2025

cp's OEIS Frontend

A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.

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