A368364 - OEIS

%I #20 Jan 09 2025 12:39:21

%S 0,1,2,4,5,7,8,11,12,14,16,18,20,23,24,26,29,31,33,36,38,40,42,45,47,

%T 51,52,54,58,60,61,65,67,71,73,75,77,81,83,85,89,91,93,98,100,102,104,

%U 107,110,114,116,118,122,125,127,131,133,135,139,141,143,149,150,154

%N a(n) = number of s with n^k-n^2 <= s <= n^k-1, k >= 3, such that a comma sequence in base n with initial term s will not reach n^k.

%C Conjectured to have g.f. (Sum_{n>=1} x^((n^2+3*n)/2)/(1-x^n) - x^2)/(1-x). [Corrected by _N. J. A. Sloane_, May 14 2024]

%C a(n) is independent of k provided k >= 3.

%C This is conjectured to equal A368363(n) - 1. Normally that would be enough to rule out this sequence. However, it is included because it is at present the only one of the nearly 100 OEIS entries based on comma sequences which has a connection with a sequence not connected with comma sequences.

%C (In the (virtual) graph that shows connections between OEIS entries, this sequence is the sole node at present that connects the component containing A121805 to the rest of the graph.)

%H Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, <a href="http://arxiv.org/abs/2401.14346">arXiv:2401.14346</a>, Fibonacci Quarterly 62:3 (2024), 215-232.

%H Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, <a href="/A121805/a121805_1.pdf">The Comma Sequence: A Simple Sequence With Bizarre Properties</a>, Local copy.

%H N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Eric Angelini's Comma Sequence</a>, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane2024.pdf">Slides</a>

%e In base 10, a(10) = 8 values of s hit a landmine before reaching safety.

%Y Cf. A121805, A136107, A368363.

%K nonn

%O 2,3

%A _N. J. A. Sloane_, Jan 19 2024