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Numbers k such that the Motzkin number A001006(k) is divisible by 5.
1

%I #11 Feb 02 2021 03:44:41

%S 9,13,23,34,38,59,63,84,88,99,109,113,134,138,148,159,163,184,188,209,

%T 213,224,234,238,249,259,263,273,284,288,309,313,334,338,349,359,363,

%U 373,384,388,398,409,413,434,438,459,463,474,484,488,509,513,523,534,538

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

%C 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).

%H Amiram Eldar, <a href="/A338220/b338220.txt">Table of n, a(n) for n = 1..10000</a>

%H Rob Burns, <a href="https://arxiv.org/abs/1611.04910">Asymptotic density of Motzkin numbers modulo small primes</a>, arXiv:1611.04910 [math.NT], 2016.

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

%t 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] &]

%Y Cf. A001006.

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

%K nonn

%O 1,1

%A _Amiram Eldar_, Jan 30 2021