A034932 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 16.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 4, 15, 6, 1, 1, 7, 5, 3, 3, 5, 7, 1, 1, 8, 12, 8, 6, 8, 12, 8, 1, 1, 9, 4, 4, 14, 14, 4, 4, 9, 1, 1, 10, 13, 8, 2, 12, 2, 8, 13, 10, 1, 1, 11, 7, 5, 10, 14, 14, 10

Offset

0,5

Comments

T(n+1,k) = (T(n,k) + T(n,k-1)) mod 16. - Reinhard Zumkeller, Mar 14 2015

Links

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)

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.

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(i, j) = binomial(i, j) mod 16.

Example

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   4  15   6   1
          1   7   5   3   3   5   7   1
        1   8  12   8   6   8  12   8   1
      1   9   4   4  14  14   4   4   9   1
    1  10  13   8   2  12   2   8  13  10   1
  1  11   7   5  10  14  14  10   5   7  11   1
.
Written in hexadecimal (with a=10, b=11, ..., f=15), rows 0..32 are
.
                                   1
                                  1 1
                                 1 2 1
                                1 3 3 1
                               1 4 6 4 1
                              1 5 a a 5 1
                             1 6 f 4 f 6 1
                            1 7 5 3 3 5 7 1
                           1 8 c 8 6 8 c 8 1
                          1 9 4 4 e e 4 4 9 1
                         1 a d 8 2 c 2 8 d a 1
                        1 b 7 5 a e e a 5 7 b 1
                       1 c 2 c f 8 c 8 f c 2 c 1
                      1 d e e b 7 4 4 7 b e e d 1
                     1 e b c 9 2 b 8 b 2 9 c b e 1
                    1 f 9 7 5 b d 3 3 d b 5 7 9 f 1
                   1 0 8 0 c 0 8 0 6 0 8 0 c 0 8 0 1
                  1 1 8 8 c c 8 8 6 6 8 8 c c 8 8 1 1
                 1 2 9 0 4 8 4 0 e c e 0 4 8 4 0 9 2 1
                1 3 b 9 4 c c 4 e a a e 4 c c 4 9 b 3 1
               1 4 e 4 d 0 8 0 2 8 4 8 2 0 8 0 d 4 e 4 1
              1 5 2 2 1 d 8 8 2 a c c a 2 8 8 d 1 2 2 5 1
             1 6 7 4 3 e 5 0 a c 6 8 6 c a 0 5 e 3 4 7 6 1
            1 7 d b 7 1 3 5 a 6 2 e e 2 6 a 5 3 1 7 b d 7 1
           1 8 4 8 2 8 4 8 f 0 8 0 c 0 8 0 f 8 4 8 2 8 4 8 1
          1 9 c c a a c c 7 f 8 8 c c 8 8 f 7 c c a a c c 9 1
         1 a 5 8 6 4 6 8 3 6 7 0 4 8 4 0 7 6 3 8 6 4 6 8 5 a 1
        1 b f d e a a e b 9 d 7 4 c c 4 7 d 9 b e a a e d f b 1
       1 c a c b 8 4 8 9 4 6 4 b 0 8 0 b 4 6 4 9 8 4 8 b c a c 1
      1 d 6 6 7 3 c c 1 d a a f b 8 8 b f a a d 1 c c 3 7 6 6 d 1
     1 e 3 c d a f 8 d e 7 4 9 a 3 0 3 a 9 4 7 e d 8 f a d c 3 e 1
    1 f 1 f 9 7 9 7 5 b 5 b d 3 d 3 3 d 3 d b 5 b 5 7 9 7 9 f 1 f 1
   1 0 0 0 8 0 0 0 c 0 0 0 8 0 0 0 6 0 0 0 8 0 0 0 c 0 0 0 8 0 0 0 1

Mathematica

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 16] (* Robert G. Wilson v, May 26 2004 *)

Prog

(Haskell)
a034932 n k = a034932_tabl !! n !! k
a034932_row n = a034932_tabl !! n
a034932_tabl = iterate
   (\ws -> zipWith ((flip mod 16 .) . (+)) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Mar 14 2015

Crossrefs

Cf. A007318, A047999, A083093, A034931, A034930, A008975.

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), (this sequence) (m = 16).

Sequence in context: A095144 A339359 A144398 * A374378 A180183 A273914

Adjacent sequences: A034929 A034930 A034931 * A034933 A034934 A034935

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved