A330509 Triangle read by rows: T(n,k) is the number of 4-ary strings of length n with k indispensable digits, with 0 <= k <= n.

1, 1, 3, 1, 9, 6, 1, 19, 34, 10, 1, 34, 115, 91, 15, 1, 55, 301, 445, 201, 21, 1, 83, 672, 1582, 1338, 392, 28, 1, 119, 1344, 4600, 6174, 3410, 700, 36, 1, 164, 2478, 11623, 22548, 19784, 7723, 1170, 45, 1, 219, 4290, 26452, 69834, 88428, 55009, 15999, 1857, 55

Offset

0,3

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 7233355548, the digits 7, 5, 5, 5, and 8 are indispensable.

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

Links

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

J. Y. Choi, Indispensable digits for digit sums, Notes Number Theory Discrete Math 25 (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) = A008287(n, 3k-2) + A008287(n, 3k-1) + A008287(n, 3k).

Example

Triangle begins
  1;
  1,   3;
  1,   9,   6;
  1,  19,  34,  10;
  1,  34, 115,  91,  15;
  1,  55, 301, 445, 201,  21;
  ...
There is 1 string (00) of length 2 with 0 indispensable digits.
There are 9 strings (01, 02, 03, 10, 12, 13, 20, 23, 30) of length 2 with 1 indispensable digit.
There are 6 strings (11, 21, 22, 31, 32, 33) of length 2 with 2 indispensable digits.
Hence T(2,0)=1, T(2,1)=9, T(2,2)=6.

Mathematica

Table[Total@ Map[Sum[Binomial[n, i] Binomial[n, # - 2 i], {i, 0, #/2}] &, 3 k + {-2, -1, 0}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 23 2019, after Jean-François Alcover at A008287 *)

Prog

(PARI) A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
T(n, k) = A008287(n, 3*k-2)+A008287(n, 3*k-1) + A008287(n, 3*k);

Crossrefs

Cf. A008287, A330381.

Sequence in context: A105539 A132819 A264364 * A105545 A178831 A027465

Adjacent sequences: A330506 A330507 A330508 * A330510 A330511 A330512

Keyword

nonn,tabl,base

Author

Ji Young Choi, Dec 16 2019

Status

approved