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<DIV>Seqfans, <BR> The magic of
the OEIS is to uncover unexpected connections </DIV>
<DIV>between seemingly unrelated methods of generating sequences. </DIV>
<DIV>Herein is just one demonstration of this magic; of course, </DIV>
<DIV>there are many other examples. </DIV>
<DIV> </DIV>
<DIV>A recent sequence submitted by Gary W. Adamson is intriguing. </DIV>
<DIV>It is entry <STRONG>A125714</STRONG> -- "Alfred Moessner's factorial
triangle" </DIV>
<DIV>I tried a variant of the generating rule, and found something </DIV>
<DIV>quite unexpected. </DIV>
<DIV> </DIV>
<DIV><EM>I encourage others to explore variations of the unusual recurrence
<BR>of A125714 and find interesting versions for themselves!</EM> </DIV>
<DIV> </DIV>
<DIV>My variation is as follows. </DIV>
<DIV> <BR>Start with a row of all 1's. Row n+1 equals the partial sums of
row n <BR>excluding terms in columns k = {1,4,8,13,...} = m*(m+1)/2 - 2 for
m>=1. </DIV>
<DIV>The terms that are not to be included in the partial sums are <BR>enclosed
in parenthesis in the table below. </DIV>
<DIV> </DIV>
<DIV>For example: <BR>row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...];
<BR>row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];<BR>row 4 =
partial sums of [1, 23,52, 141,216,306, 535,695,876,1079,
...].<BR> <BR>Rows of table begin: <BR>1,(<STRONG>1</STRONG>), 1,
1,(<STRONG>1</STRONG>), 1, 1, 1,(<STRONG>1</STRONG>), 1, 1, 1,
1,(<STRONG>1</STRONG>), 1, 1, 1, ...;<BR>1,(<STRONG>2</STRONG>), 3,
4,(<STRONG>5</STRONG>), 6, 7, 8,(<STRONG>9</STRONG>), 10, 11, 12,
13,(<STRONG>14</STRONG>), 15, 16, 17, ...;<BR>1,(<STRONG>4</STRONG>), 8,
14,(<STRONG>21</STRONG>), 29, 39, 50,(<STRONG>62</STRONG>), 75, 90, 106,
123,(<STRONG>141</STRONG>), 160, 181,.;<BR>1,(<STRONG>9</STRONG>), 23,
52,(<STRONG>91</STRONG>), 141, 216, 306,(<STRONG>412</STRONG>), 535, 695, 876,
1079,(<STRONG>1305</STRONG>),..;<BR>1,(<STRONG>24</STRONG>), 76,
217,(<STRONG>433</STRONG>), 739, 1274, 1969,(<STRONG>2845</STRONG>), 3924, 5479,
7335,...;<BR>1,(<STRONG>77</STRONG>), 294, 1033,(<STRONG>2307</STRONG>), 4276,
8200, 13679,(<STRONG>21014</STRONG>), 30534,
45528,...;<BR>1,(<STRONG>295</STRONG>), 1328, 5604,(<STRONG>13804</STRONG>),
27483, 58017, 103545,(<STRONG>167868</STRONG>),
255305,...;<BR>1,(<STRONG>1329</STRONG>), 6933, 34416,(<STRONG>92433</STRONG>),
195978, 451283,
855463,(<STRONG>1454823</STRONG>),...;<BR>1,(<STRONG>6934</STRONG>), 41350,
237328,(<STRONG>688611</STRONG>), 1544074, 3847960,
7700971,...;<BR>1,(<STRONG>41351</STRONG>), 278679,
1822753,(<STRONG>5670713</STRONG>), 13371684, 35818351,
75299744,...;<BR>...</DIV>
<DIV> </DIV>
<DIV>The amazing connection here is this: </DIV>
<DIV><BR>column 1 equals A091352 = column 1 of triangle A091351, <BR>in which
column k equals row sums of the matrix power A091351^k. </DIV>
<DIV> </DIV>
<DIV>I do not have a proof, but (at least) <EM>the
initial <STRONG>50</STRONG> terms match exactly</EM>. </DIV>
<DIV> </DIV>
<DIV>Triangle A091351 begins:</DIV>
<DIV>1; <BR>1, <STRONG>1</STRONG>; <BR>1, <STRONG>2</STRONG>,
1; <BR>1, <STRONG>4</STRONG>, 3, 1; <BR>1, <STRONG>9</STRONG>, 9, 4,
1; <BR>1, <STRONG>24</STRONG>, 30, 16, 5, 1; <BR>1,
<STRONG>77</STRONG>, 115, 70, 25, 6, 1; <BR>1, <STRONG>295</STRONG>, 510,
344, 135, 36, 7, 1; <BR>1, <STRONG>1329</STRONG>, 2602, 1908, 805, 231,
49, 8, 1; <BR>1, <STRONG>6934</STRONG>, 15133, 11904, 5325, 1616, 364, 64,
9, 1; ... </DIV>
<DIV> </DIV>
<DIV>where <EM>column <STRONG>k</STRONG> of
<STRONG>A091351</STRONG> = row sums of matrix
power <STRONG>A091351</STRONG>^<STRONG>k</STRONG></EM> for k>=0. </DIV>
<DIV> </DIV>
<DIV>Quite an unexepected connection between such diverse recurrences!
</DIV>
<DIV> Paul </DIV>
<DIV> </DIV></BODY></HTML>