Riffs & Rotes

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

R&R.  Note 24

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

A109300 [pending]

%I A109300
%S A109300 1, 1, 7, 999999991
%N A109300 a(n) = number of positive integers whose rote height in gammas is n.
%C A109300 A109300 is the sequence of first differences of A050924.  Conversely,
           A050924 is the sequence of running subtotals of A109300.  This can be
           seen as follows.  Let P(0) c P(1) c ... c P(n) c ... be an increasing
           sequence of sets of partial functions that is defined by the recursion:
           P(0) = {the empty function}, P(n+1) = {partial functions: P(n) -> P(n)}.
           Then |P(n)| = A050924(n+1) = number of positive integers of rote height
           at most n, hence |P(n)| - |P(n-1)| = A109300(n) = number of positive
           integers of rote height exactly n.
%e A109300 Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
%e A109300 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A109300 o-o ` ` o-o ` ` ` o-o ` o-o o-o ` ` o-o o-o ` ` ` o-o o-o ` ` o-o o-o o-o
%e A109300 | ` ` ` | ` ` ` ` | ` ` | ` | ` ` ` | ` | ` ` ` ` | ` | ` ` ` | ` | ` | `
%e A109300 o-o ` o-o ` ` o-o o-o ` o---o ` ` o-o ` o-o ` o-o o---o ` ` o-o ` o---o `
%e A109300 | ` ` | ` ` ` | ` | ` ` | ` ` ` ` | ` ` | ` ` | ` | ` ` ` ` | ` ` | ` ` `
%e A109300 O ` ` O ` ` ` O===O ` ` O ` ` ` ` O=====O ` ` O===O ` ` ` ` O=====O ` ` `
%e A109300 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A109300 2:1 ` 1:2 ` ` 1:1 2:1 ` 2:2 ` ` ` 1:2 2:1 ` ` 1:1 2:2 ` ` ` 1:2 2:2 ` ` `
%e A109300 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%e A109300 3 ` ` 4 ` ` ` 6 ` ` ` ` 9 ` ` ` ` 12` ` ` ` ` 18` ` ` ` ` ` 36` ` ` ` ` `
%e A109300 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
%Y A109300 Cf. A007097, A050924, A061396, A062504, A062537, A062860, A109301.
%K A109300 nonn,new
%O A109300 0
%A A109300 Jon Awbrey (jawbrey(AT)att.net), Jul 04 2005

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o