A338220 Numbers k such that the Motzkin number A001006(k) is divisible by 5.

9, 13, 23, 34, 38, 59, 63, 84, 88, 99, 109, 113, 134, 138, 148, 159, 163, 184, 188, 209, 213, 224, 234, 238, 249, 259, 263, 273, 284, 288, 309, 313, 334, 338, 349, 359, 363, 373, 384, 388, 398, 409, 413, 434, 438, 459, 463, 474, 484, 488, 509, 513, 523, 534, 538

Offset

1,1

Comments

The asymptotic density of this sequence is 1/10. It is a disjoint union of 4 sequences: numbers of the form (5*i + 1)*5^(2*j) - 2, (5*i + 2)*5^(2*j-1) - 1, (5*i + 3)*5^(2*j-1) - 2, and (5*i + 4)*5^(2*j) - 1, with i>=0 and j>=1, whose asymptotic densities are 1/120, 1/24, 1/24, and 1/120, respectively (Burns, 2016).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Example

9 is a term since A001006(9) = 835 = 5 * 167 is divisible by 5.

Mathematica

motz[0] = motz[1] = 1; motz[n_] := motz[n] = ((2*n + 1)*motz[n - 1] + 3*(n - 1)*motz[n - 2])/(n + 2);  Select[Range[0, 500], Divisible[motz[#], 5] &]

Crossrefs

Cf. A001006.

Similar sequences, indices of Motzkin numbers divisible by m: A081706 (m = 2), A089119 (m = 3).

Sequence in context: A151901 A157199 A291351 * A242919 A097539 A107913

Adjacent sequences: A338217 A338218 A338219 * A338221 A338222 A338223

Keyword

nonn

Author

Amiram Eldar, Jan 30 2021

Status

approved