No subject

Sat May 12 00:50:11 CEST 2001

The following is being sent to seq-fans in
the hope that at least one will check it and turn
it into an acceptable form for Neil Sloane.  He has
already included the `squares' and `triangles'
sequences, I believe.      R.

Here are some `add the greatest whatever' sequences.

They arose from `put-or-take' games (see Winning Ways,
484--486, 501--503) the prototype being Epstein's
Put-or-Take-a-Square game.

Start with a heap of 1 (or more?) and successively add
the largest member of your favorite sequence which is
not greater (strictly less?) than the present size of
the heap.

With squares we get

%1 = 1  %2 = 2  %3 = 3  %4 = 4  %5 = 8  %6 = 12  %7 = 21
%8 = 37  %9 = 73  %10 = 137  %11 = 258  %12 = 514
%13 = 998  %14 = 1959  %15 = 3895  %16 = 7739
%17 = 15308  %18 = 30437  %19 = 60713  %20 = 121229
%21 = 242333  %22 = 484397  %23 = 967422  %24 = 1933711
%25 = 3865811  %26 = 7730967  %27 = 15459367
%28 = 30912128  %29 = 61814609  %30 = 123625653
%31 = 247235577  %32 = 494448306  %33 = 988888002
%34 = 1977738918  %35 = 3955408759  %36 = 7910812423
%37 = 15821491787  %38 = 31642854876  %39 = 63285572332
%40 = 126571024688  %41 = 253141894512
%42 = 506283703936  %43 = 1012567183232
%44 = 2025132408401

With `squares + 1' we get

1  2  4  6  11  21  38  75  140  262  519  1004  1966
3903  7748  15493  30870  61496  122506  245007
489044  977646  1953791  3905401  7809978  15616415
31226817  62452562  124894167  249774793  499541210
999063711  1998066161  3996066763  7992076560
15984078965  31968118150  63936127767  127871778793
255743102075  511485706176  1022970999301
2045941415702  4091882588199  8183764253800
16367523222330

If we add triangular numbers (as in `Tribulations') we get

1  2  3  6  12  22  43  79  157  310  610  1205
2381  4727  9383  18699  37227  74355  148660
296900  593735  1187240  2373810  4746741  9491481
18981027  37956907  75910735  151820416  303627016
607253419  1214497244  2428978214  4857918665
9715793261  19431485367  38862885183  77725514211
155450916339  310901711244  621802925484
1243605547519  2487209615935  4974418988905
9948836839658

`Triangular numbers + 1' (are never multiples of 3,
and hence yield a game which *may* be feasible to
analyze).  If we add these we get

1  2  4  8  15  26  48  94  186  358  710  1414
2793  5569  11135  22162  44318  88572  176983
353699  706920  1413187  2825228  5649105
11295586  22588715  45171276  90339037  180675999
361337536  722671940  1445337094  2890647825
5781270421  11562459578  23124873925

If you do it with the powers of 2, you just get powers
of 2.  Let's try with powers of 3:

1  2  3  6  9  18  27  54  81  162  243  486  729  ...

(not in Ency Seq, but not very exciting).

If you use the odd numbers you get M0788 (A5578).

The Fibonacci numbers give you

1  2  4  7  12  20  33  54  88  143  232  ...

i.e., the Fibs - 1  M1056, A0071, N0397.

Of course, we can use any sequence in the database,
and we can iterate.  E.g., if I now use the Fibs - 1
I get

1  2  4  8  15  27  47  80  134  222  ...

which are  M1103  A0126  N0421,  the Fibs minus n-1
-- I haven't checked  FQ 3 295 65  to see if that
manifestation is listed.  Continuing to iterate
gives

1  2  4  8  16  31  58  105  185  319  ...

which are  M1120  A0128  N0428.  They are also the
Fibs - (n^2-7n+18)/2 in case anyone is interested.

If we push it one stage further, then we seem to be
out of Ency Seq -- are we out of the database?

1  2  4  8  16  32  63  121  226  411  730 ...

I think that they are  Fibs - (n^3-15n^2+92n-186)/6

Now let's use the  Fibs + 1  as in Mike Guy's game
of Fibulations, evidently  M0574  A0381  N1692,
though Ency Seq doesn't give this simple definition.
We now get:

1  2  4  8  14  28  50  95  185  330  564  942 ...

and when you try to relate these to the Fibs it
seems that they differ by a step function with
increasingly long steps.

If you use the numbers  2^k - 1, the game is
She-Loves-Me-She-Loves-Me-Not, and the sequence is

1  2  3  6  9  16  31  62  93  156  283  538  ...

(not in the database ??) and you get a similar
`seven-league boots' phenomenon: when you drop
below a power of 2, e.g at 31, then the next
33 terms are  32+30, 64+29, 128+28, 256+27,
... 2^34 + 1, 2^35, 2^36 - 1, 2^37 - 2, and now
you may only add  2^36 - 1 (again), giving
2^37 + (2^36 - 3)  and the pattern goes on
for about 2^36 terms before donning our next
size of boot -- a bit bigger than 2^(2^36) !

What is of interest when calculating remotenesses
and suspense numbers is the remainder when
you *subtract* these numbers from the heap.
E.g.. if we take the largest square from the
members of the `squares' sequence that we started
with, we get

0  1  2  0  4  3  5  1  9  16  2  30  37  23  51 ...

Now if anyone could describe this sequence (in
some other way!) we might be nearer to analyzing
the game.  In the case of Fibulations, this can
be done with the second cousin of the Zeckendorf
representation, called Secondoff in WW.  Also,
in the  2^k - 1  example, it can be done in
terms of the binary representation.  Problem:
invent a numeration scheme to describe each of
these examples.

Meanwhile, we could use the pentagonal numbers of
positive rank  M3818  A0326  N1562  to give

1  2  3  4  5  10  15  27  49  84  154  299
586  1118  2198  ...

or the pentagonal numbers of any rank  M1336
A1318  N0511  to give

1  2  4  6  11  18  33  59  116  216  426  851 ...

Neither in Ency seq, though M1018--1020 are close
to the latter.

Before I close, I must try the Catalan numbers:

1  2  4  6  11  16  30  44  86  128  170  302
434  863  1292  1721  3151  4581  6011  10873
15735  20597  37393  54189  70985  129771  ...

and the primes (purists will start at 2)

1  2  4  7  14  27  50  97  194  387  770  1539
3070  6137  12270  24539  49072  98141  196270 ...

For `squares minus one' you need to start at 3:

3  6  9  17  32  56  104  203  398  758  1486
2929  5844  11620  23068  45868  ... 

Feel free to add, and iterate, and vary, to taste. R.