A340620 T(n,k) is the number of 4-ary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit with 0 <= k <= n.

3, 6, 6, 10, 28, 10, 15, 81, 81, 15, 21, 186, 354, 186, 21, 28, 371, 1137, 1137, 371, 28, 36, 672, 3018, 4836, 3018, 672, 36, 45, 1134, 7023, 16374, 16374, 7023, 1134, 45, 55, 1812, 14829, 47286, 68644, 47286, 14829, 1812, 55, 66, 2772, 29043, 121314, 240021, 240021, 121314, 29043, 2772, 66

Offset

0,1

Comments

A digit in a string is called indispensable if it is greater than the following digit or equal to the following digits which are eventually greater than the following digit. We also assume that there is an invisible digit 0 at the end of any string. For example, in the string 33102232, the digits 3, 3, 1, 3, and 2 are indispensable (from the left).

T(n,k) is also the number of integers m where the length of base-4 representation of m is n+k and the digit sum of the base-4 representation of 3m is 3(k+1).

Links

Table of n, a(n) for n=0..54.

J. Y. Choi, Indispensable digits for digit sums, Notes on Number Theory and Discrete Mathematics, 25(2), (2019), pp. 40-48.

J. Y. Choi, Digit sums generalizing binomial coefficients, J. Integer Seq. 22 (2019), Article 19.8.3.

Formula

T(n,k) = A330509(n+1,k+1) - A330509(n,k+1).

Example

Triangle begins
   3;
   6,   6;
  10,  28,   10;
  15,  81,   81,   15;
  21, 186,  354,  186,   21;
  28, 371, 1137, 1137,  371,  28;
  36, 672, 3018, 4836, 3018, 672, 36;
  ...
There are 6 4-ary strings (10, 12, 13, 20, 23, 30) of length 2 with 1 indispensable digits and a nonzero leading digit.
There are 6 4-ary strings (11, 21, 22, 31, 32, 33) of length 2 with 2 indispensable digits and a nonzero leading digit.
There are 10 4-ary strings (111, 211, 221, 222, 311, 321, 322, 331, 332, 333) of length 3 with 3 indispensable digits and a nonzero leading digit.
Hence, T(1,0)=6, T(1,1)=6, T(2,2)=10.

Prog

(PARI) A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
A330509(n, k) = A008287(n, 3*k-2)+A008287(n, 3*k-1) + A008287(n, 3*k);
T(n, k) = A330509(n+1, k+1) - A330509(n, k+1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jan 19 2021

Crossrefs

Cf. A330381, A330509, A330510, A005773, A008287.

Sequence in context: A332546 A002853 A278807 * A184137 A135610 A157018

Adjacent sequences: A340617 A340618 A340619 * A340621 A340622 A340623

Keyword

nonn,base

Author

Ji Young Choi, Jan 13 2021

Extensions

More terms from Michel Marcus, Jan 19 2021

Status

approved