A282715 Number of nonzero entries in row n of the base-3 generalized Pascal triangle P_3.

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

Offset

0,2

Comments

It would be nice to have an entry for the triangle P_3 itself (compare A282714 which gives the base-2 triangle P_2).

Links

Lars Blomberg, Table of n, a(n) for n = 0..10000

Julien Leroy, Michel Rigo, Manon Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics 340 (2017), 862-881, Section 7.

Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.

Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.

Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.

Formula

Leroy et al. (2017) state some conjectured recurrences.

Example

The number of nonzero entries in the n-th row of the following triangle:
1
1 1
1 0 1
1 1 0 1
1 2 0 0 1
1 1 1 0 0 1
1 0 1 0 0 0 1
1 1 1 0 0 0 0 1
1 0 2 0 0 0 0 0 1
1 1 0 2 0 0 0 0 0 1
1 2 0 1 1 0 0 0 0 0 1
1 1 1 1 0 1 0 0 0 0 0 1
1 2 0 2 1 0 0 0 0 0 0 0 1
1 3 0 0 3 0 0 0 0 0 0 0 0 1

Maple

# reuses code snippets of A282714
A282715 := proc(n)
    add(min(P(n, k, 3), 1), k=0..n) ;
end proc:
seq(A282715(n), n=0..100) ; # R. J. Mathar, Mar 03 2017

Mathematica

row[n_] := Module[{bb, ss}, bb = Table[IntegerDigits[k, 3], {k, 0, n}]; ss = Subsets[Last[bb]]; Prepend[Count[ss, #]& /@ bb // Rest, 1]];
a[n_] := Count[row[n], _?Positive];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 28 2017 *)

Crossrefs

Cf. A007306, A282714.

Sequence in context: A269989 A057935 A292042 * A124831 A105096 A157790

Adjacent sequences: A282712 A282713 A282714 * A282716 A282717 A282718

Keyword

nonn

Author

N. J. A. Sloane, Mar 02 2017

Extensions

More terms from Lars Blomberg, Mar 03 2017

Status

approved