A380834 First column of Kimberling's ESC array.

1, 4, 7, 9, 11, 15, 17, 20, 22, 25, 27, 30, 33, 35, 38, 41, 43, 46, 49, 51, 53, 57, 59, 61, 64, 67, 69, 72, 75, 77, 79, 83, 85, 88, 90, 93, 95, 99, 101, 103, 106, 109, 111, 114, 117, 119, 121, 125, 127, 130, 132, 135, 137, 140, 143, 145, 148, 151, 153, 156

Offset

1,2

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).

There is an 18-state Fibonacci automaton (see escm1) that, on input n in Zeckendorf representation, accepts n if and only if it belongs to this sequence. Proved with the Walnut theorem prover.

Links

Table of n, a(n) for n=1..60.

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, escm1 automaton.

Crossrefs

Cf. A380835.

Sequence in context: A025054 A310947 A275834 * A310948 A310949 A139586

Adjacent sequences: A380831 A380832 A380833 * A380835 A380836 A380837

Keyword

nonn,new

Author

Jeffrey Shallit, Feb 05 2025

Status

approved