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

%I A282715 #33 Feb 24 2025 03:03:19
%S A282715 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,
%T A282715 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,
%U A282715 8,10,7,10,12,13,11,8,11,13,12,10,7,10,8,11,9,10
%N A282715 Number of nonzero entries in row n of the base-3 generalized Pascal triangle P_3.
%C A282715 It would be nice to have an entry for the triangle P_3 itself (compare A282714 which gives the base-2 triangle P_2).
%H A282715 Lars Blomberg, <a href="/A282715/b282715.txt">Table of n, a(n) for n = 0..10000</a>
%H A282715 Julien Leroy, Michel Rigo, and Manon Stipulanti, <a href="http://dx.doi.org/10.1016/j.disc.2017.01.003">Counting the number of non-zero coefficients in rows of generalized Pascal triangles</a>, Discrete Mathematics 340 (2017), 862-881, Section 7.
%H A282715 Julien Leroy, Michel Rigo, and Manon Stipulanti, <a href="https://arxiv.org/abs/1705.10065">Counting Subwords Occurrences in Base-b Expansions</a>, arXiv:1705.10065 [math.CO], 2017.
%H A282715 Julien Leroy, Michel Rigo, and Manon Stipulanti, <a href="http://math.colgate.edu/~integers/sjs13/sjs13.Abstract.html">Counting Subwords Occurrences in Base-b Expansions</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
%H A282715 Manon Stipulanti, <a href="https://arxiv.org/abs/1801.03287">Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems</a>, arXiv:1801.03287 [math.CO], 2018.
%F A282715 Leroy et al. (2017) state some conjectured recurrences.
%e A282715 The number of nonzero entries in the n-th row of the following triangle:
%e A282715   1
%e A282715   1 1
%e A282715   1 0 1
%e A282715   1 1 0 1
%e A282715   1 2 0 0 1
%e A282715   1 1 1 0 0 1
%e A282715   1 0 1 0 0 0 1
%e A282715   1 1 1 0 0 0 0 1
%e A282715   1 0 2 0 0 0 0 0 1
%e A282715   1 1 0 2 0 0 0 0 0 1
%e A282715   1 2 0 1 1 0 0 0 0 0 1
%e A282715   1 1 1 1 0 1 0 0 0 0 0 1
%e A282715   1 2 0 2 1 0 0 0 0 0 0 0 1
%e A282715   1 3 0 0 3 0 0 0 0 0 0 0 0 1
%p A282715 # reuses code snippets of A282714
%p A282715 A282715 := proc(n)
%p A282715     add(min(P(n,k,3),1),k=0..n) ;
%p A282715 end proc:
%p A282715 seq(A282715(n),n=0..100) ; # _R. J. Mathar_, Mar 03 2017
%t A282715 row[n_] := Module[{bb, ss}, bb = Table[IntegerDigits[k, 3], {k, 0, n}]; ss = Subsets[Last[bb]]; Prepend[Count[ss, #]& /@ bb // Rest, 1]];
%t A282715 a[n_] := Count[row[n], _?Positive];
%t A282715 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Dec 28 2017 *)
%Y A282715 Cf. A007306, A282714.
%K A282715 nonn
%O A282715 0,2
%A A282715 _N. J. A. Sloane_, Mar 02 2017
%E A282715 More terms from _Lars Blomberg_, Mar 03 2017