Riffs & Rotes & A061396

[Jon Awbrey]

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David W. Wilson wrote:

JA: Re:

    ...

    Table 3.  Triangle in which k-th row lists natural number
              values for the collection of riffs with k nodes:
    --o------------------------------------------------------------------------------
    k | natural numbers n such that |riff(n)| = k
    --o------------------------------------------------------------------------------
    0 |    1;
    1 |    2;
    2 |    3,    4;
    3 |    5,    6,    7,    8,     9,    16;
    4 |   10,   11,   12,   13,    14,    17,     18,     19,     23,     25,
      |   27,   32,   49,   53,    64,    81,    128,    256,    512,  65536;
    5 |   15,   20,   21,   22,    24,    26,     28,     29,     31,     34,
      |   36,   37,   38,   41,    43,    46,     48,     50,     54,     59,
      |   61,   67,   83,   97,    98,   103,    106,    121,    125,    131,
      |  162,  169,  227,  241,   243,   289,    311,    343,    361,    419,
      |  529,  625,  719,  729,  1024,  1619,   2048,   2187,   2401,   2809,
      | 3671, 4096, 6561, 8192, 16384, 19683, 131072, 262144, 524288, 821641,
      | 8388608,  33554432,  43046721,  134217728,  4294967296,  562949953421312,
      | 9007199254740992,  18446744073709551616,  2417851639229258349412352,
      | 2^128,  2^256,  2^512,  2^65536;
      |
    --o------------------------------------------------------------------------------

JA: 30 = p^1 p^2 p^3 = p.p< .p<
                           p   p<
                                 p

I meant, of course, to write:

    30 = p_1 p_2 p_3 = p.p< .p<
                           p   p<
                                 p

JA: => |riff(30)| = 6

    and 30 is the least number not in the above table,
    so we have Min Seq = 1, 2, 3, 5, 10, 15, 30, ...
    which is not in EIS.

    Unless I have made the classic mistake of staying up too late again.

    Which I will remedy, post haste.

DW: I was holding back these sequences, but Jon has forced my hand ...

Sorry, David, I am new here, and had no clue that
some brand of poker was being played.  I was just
treating it like all of my other discussion groups,
where I am also much too chatty for anybody's good,
but where nothing much is at stake but time, all too
precious time.  I think that I indicated in my initial
note that this is no longer my active arena, and my whole
reason for being here is turn this old hobby horse of mine
over to abler cowhands, of whose ability I already have more
than ample evidence.  So, count the ways, and do as you will!
As long as I can keep it recreational, I will skulk about the
old corral, and kibbitz on what all's going on, but I already
had my fill of this damned ole rodeo in my own wrangler days.
But I would, for a little while, like to try and comprehend
what is sprung from these seeds, so if you could explain
what is meant by a "core sequence", I would like that.

DW: I proffer this as the "core sequence" for riffs.
| 
| %I A000000
| %S A000000  0,1,2,2,3,3,3,3,3,4,4,4,4,4,5,3,4,4,4,5,5,5,4,5,4,5,4,5,5,6,5,4,6,5,6,5,
| %T A000000  5,5,6,6,5,6,5,6,6,5,6,5,4,5,6,6,4,5,7,6,6,6,5,7,5,6,6,4,7,7,5,6,6,7,6,6,
| %U A000000  6,6,6,6,7,7,6,6,4,6,5,7,7,6,7,7,6,7,7,6,7,7,7,6,5,5,7,6,6,7,5,7,8,5,6,6
| %N A000000  Nodes in riff (rooted index-functional forest) for n.
| %F A000000  a(PROD(p_i^e_i)) = SUM(a(i)+a(e_i)+1), product over nonzero e_i in
|             prime factorization.
| %H A000000  J. Awbrey, <a> href="http://www.research.att.com/~njas/sequences/a061396a.txt">
|             Illustrations of riffs for small integers </a>
| %C A000000  A061396(n) gives number of times n appears in this sequence.
| %K A000000  nonn,easy
| %O A000000  1,3
| %A A000000  dww

DW: Here is Jon's sequence extended somewhat.
    I am still looking to compute this more quickly.

DW:
| %I A000001
| %S A000001 1,2,3,5,10,15,30,55,105,165,330,660,1155,2145,4290,7755,15015,30030,
| %T A000001 54285,100815,201630,403260,705705
| %N A000001 Smallest k with n nodes in its riff (rooted index-functional forest)
|            for n.
| %K A000001 nonn
| %O A000001 0,2
| %C A000001 Smallest k with A000000(k) = n.
| %A A000001 Jon Awbrey (jawbrey at oakland.edu), dww

Thanks for the new numbers!

Many Regards,

Jon Awbrey

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