A380835 - OEIS

%I #18 Feb 06 2025 09:26:04

%S 2,6,12,14,18,24,28,32,36,40,44,48,54,56,62,66,70,74,80,82,86,92,96,

%T 98,104,108,112,116,122,124,128,134,138,142,146,150,154,160,164,166,

%U 172,176,180,184,190,192,196,202,206,210,214,218,222,226,232,234,240

%N Second column of Kimberling's ESC array.

%C Kimberling's ESC array is defined as follows: it is indexed by rows i and columns j, both from 1 to infinity. Rows follow a generalized Fibonacci sequence: ESC[i,j] = ESC[i,j-1] + ESC[i,j-2] for j>=3. The second column is defined by E[i,2] = floor(tau*E[i,1]) + (i mod 2), where tau = (1+sqrt(5))/2 is the golden ratio. Finally, the first column E[i,1] is defined to be the minimal excluded element over all previous rows (that is, the smallest positive integer not contained in those rows).

%C Kimberling conjectured that every term of this sequence is even. This was proved by Behrend (2012).

%C There is a 47-state Fibonacci automaton that, on input n in Zeckendorf representation, accepts if and only if n belongs to this sequence. Constructed with the Walnut theorem prover. See the file escm2.

%H M. Behrend, <a href="https://www.fq.math.ca/Papers1/50-2/Behrend.pdf">Proof of Kimberling's "even second column" conjecture</a>, Fibonacci Quart. 50 (2012), 106--118.

%H Clark Kimberling, <a href="https://www.mathstat.dal.ca/FQ/Scanned/32-4/kimberling.pdf">The first column of a Stolarsky interspersion</a>, Fib. Quart. 32 (1994), 301-315.

%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2502.03312">An 'experimental mathematics' approach to Stolarsky interspersions via automata theory</a>, ArXiv preprint arXiv:2502.03312 [cs.FL], February 5 2025.

%H Jeffrey Shallit, <a href="/A380835/a380835.pdf">escm2 automaton</a>.

%F a(n) = floor(tau*A380834(n)) + (n mod 2).

%Y Cf. A380834.

%K nonn,new

%O 1,1

%A _Jeffrey Shallit_, Feb 05 2025