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A338220
Numbers k such that the Motzkin number A001006(k) is divisible by 5.
1
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 30 2021
STATUS
approved