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

Author

Amiram Eldar, Jan 30 2021

Keywords

• nonn

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).

Examples

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

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.

Crossrefs

Cf. A001006.

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

Programs

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] &]