A380835 - OEIS

A380835

Second column of Kimberling's ESC array.

2, 6, 12, 14, 18, 24, 28, 32, 36, 40, 44, 48, 54, 56, 62, 66, 70, 74, 80, 82, 86, 92, 96, 98, 104, 108, 112, 116, 122, 124, 128, 134, 138, 142, 146, 150, 154, 160, 164, 166, 172, 176, 180, 184, 190, 192, 196, 202, 206, 210, 214, 218, 222, 226, 232, 234, 240

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OFFSET

1,1

COMMENTS

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).

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

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.

LINKS

Table of n, a(n) for n=1..57.

M. Behrend, Proof of Kimberling's "even second column" conjecture, Fibonacci Quart. 50 (2012), 106--118.

Clark Kimberling, The first column of a Stolarsky interspersion, Fib. Quart. 32 (1994), 301-315.

Jeffrey Shallit, An 'experimental mathematics' approach to Stolarsky interspersions via automata theory, ArXiv preprint arXiv:2502.03312 [cs.FL], February 5 2025.

Jeffrey Shallit, escm2 automaton.

FORMULA

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

CROSSREFS

Cf. A380834.

Sequence in context: A154712 A079849 A260907 * A333833 A057895 A111369

Adjacent sequences: A380832 A380833 A380834 * A380836 A380837 A380838