OFFSET
0,4
COMMENTS
In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
. Column k of
. this square
. Generalized Triangle Triangle array A195825
k m m-gonal "A" "B" [row sums of
. numbers triangle "B"
. with a(0)=1]
========================================================
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012
LINKS
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem
FORMULA
Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...
7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...
11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...
15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...
22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...
30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...
42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...
56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...
77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...
101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...
135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...
176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...
...
Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11, ... The column contains only one plateau: [1, 1] which has level 1 and length 2.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21, ... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.
PROG
(GW-BASIC)' A program (with two A-numbers) for the table of example section.
20 FOR k = 1 TO 10 'Column 1-10
30 T(0, k) = 1 'Row 0
40 FOR n = 1 TO 15 'Rows 1-15
50 FOR j = 1 TO n
70 NEXT j
80 NEXT n
90 NEXT k
100 FOR n = 0 TO 15
110 FOR j = 1 TO 10
120 PRINT T(n, k);
130 NEXT k
140 PRINT
150 NEXT n
160 END
170 'Omar E. Pol, Jun 18 2012
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Sep 24 2011
STATUS
approved