tabular sequences
Michael Somos
somos at grail.cba.csuohio.edu
Wed May 3 01:23:15 CEST 2000
Seqfans,
In order to test the limits of the On-Line EIS, I propose the
following two examples as possible sequence entries.
The first is based directly on Table 10.4 on page 87 of Ross
Honsberger's "Ingenuity in Mathematics" which looks like this
( when rotated a quarter-turn clockwise )
-------------------------------------------------------------------------------
1
4 3 2
9 8 7 6 5
16 15 14 13 12 11 10
25 24 23 22 21 20 19 18 17
36 35 34 33 32 31 30 29 28 27 26
49 48 47 46 45 44 43 42 41 40 39 38 37
64 63 62 61 60 59 58 57 56 55 54 53 52 51 50
-------------------------------------------------------------------------------
Note the central column might yield the explanation of the name "Central
polygonal numbers" of (A002061), but that may be given in Hogben.
===============================================================================
%I A002061 M2638 N1049
%S A002061 1,1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,
%T A002061 421,463,507,553,601,651,703,757,813,871,931,993,1057,1123,1191,1261
%N A002061 Central polygonal numbers: n^2 - n + 1.
%D A002061 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1,
Chanticleer Press, NY, 1950, p. 22.
%D A002061 R. Honsberger, Ingenuity in Math., Random House, 1970, p. 87.
%O A002061 0,3
%A A002061 njas
%K A002061 nonn,easy,nice
===============================================================================
Note the first element of each row is the squares (A000290) while the
last element of each row is one more than squares (A002522) and the
center element of each row is not named in Honsberger but is (A002061).
The column to the left of the center column is (A014206). Just left of
that is (A027689), and so on. Working to the right, the column to the
right of center is the heteromecic numbers (A002378). To the right of
that is (A028387). Next is (A028552). Next is (A014209) and no more in
EIS. Note the row sums are the centered cube numbers (A005898).
How should this sequence table be listed? It looks like it should be :
-------------------------------------------------------------------------------
%S A?????? 1,4,3,2,9,8,7,6,5,16,15,14,13,12,11,10,25,24,23,22,21,20,19,18,17,36,
%T A?????? 35,34,33,32,31,30,29,28,27,26,49,48,47,46,45,44,43,42,41,40,39,38,37,
%U A?????? 64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,81,80,79,78,77,76,75,74
-------------------------------------------------------------------------------
However, suppose we prefer to reverse the order of entries in each row. Then :
-------------------------------------------------------------------------------
%S A000027 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T A000027 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,
%U A000027 48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69
-------------------------------------------------------------------------------
is an already known sequence, but it is not regarded as a table! Is it possible
to have a sequence of numbers which is read in more than one way?
Suppose we consider a closely related and simpler case of the following :
-------------------------------------------------------------------------------
1
3 2
6 5 4
10 9 8 7
15 14 13 12 11
21 20 19 18 17 16
28 27 26 25 24 23 22
36 35 34 33 32 31 30 29
45 44 43 42 41 40 39 38 37
55 54 53 52 51 50 49 48 47 46
66 65 64 63 62 61 60 59 58 57 56
-------------------------------------------------------------------------------
Note the first element of each row is the triangular numbers (A000217) while
the last element of each row is one more than the triangular numbers (A000124)
yet is also named as the central polygonal numbers in EIS. The center column
is known as the centered square numbers (A001844). To the left of the center
column is 3,9,19,33,51,... which seems not to be in EIS. To the left of that
is 6,14,26,42,... which is (A051890) and recently mentioned here in Seqfan.
To the right of center is 2,8,18,32,50,... which are nuclear "magic numbers"
(A001105). To the right of that is 4,12,24,40,60,... which is (A046092). To
the right of that is 7,17,31,49,71,... which is 2^n^2-1 and seems not to be
in EIS. The rows sums are (A006003) which has a comment about this from Russo.
How should this sequence table be listed? It looks like it should be :
-------------------------------------------------------------------------------
%S A?????? 1,3,2,6,5,4,10,9,8,7,15,14,13,12,11,21,20,19,18,17,16,28,27,26,25,24,
%T A?????? 23,22,36,35,34,33,32,31,30,29,45,44,43,42,41,40,39,38,37,55,54,53,52,
%U A?????? 51,50,49,58,47,46,66,65,64,63,62,61,60,59,58,57,56,78,77,76,75,74,73
-------------------------------------------------------------------------------
However, again, if we prefer to reverse the order of entries of each row we
get A000027 again with a different reading!
--
Michael Somos <somos at grail.cba.csuohio.edu> Cleveland State University
http://grail.cba.csuohio.edu/~somos/ Cleveland, Ohio, USA 44115
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