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Search: id:A101321
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| A101321 |
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Table read by anti-diagonals: row m gives the sequence of centered figurate numbers of the m-gon. |
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3
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| 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67
(list; table; graph; listen)
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OFFSET
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0,5
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FORMULA
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1+m*(n+n^2)/2
n-th row of the array = A001263 * [1, n, 0, 0, 0,...] where A001263 = the Narayana triangle. E.g. A001263 * [1, 2, 0, 0, 0,...] = (1, 3, 7, 13, 21, 31, 43,...) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 30 2007
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EXAMPLE
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The upper left 10 X 10 corner of the infinite table T is
|0|1 1 1 1 1 1 1 1 1 1|
|1|1 2 4 7 11 16 22 29 37 46|
|2|1 3 7 13 21 31 43 57 73 91|
|3|1 4 10 19 31 46 64 85 109 136|
|4|1 5 13 25 41 61 85 113 145 181|
|5|1 6 16 31 51 76 106 141 181 226|
|6|1 7 19 37 61 91 127 169 217 271|
|7|1 8 22 43 71 106 148 197 253 316|
|8|1 9 25 49 81 121 169 225 289 361|
|9|1 10 28 55 91 136 190 253 325 406|
Item 11 of row 5 is 331 because we can write 1+5*(11+11^2)/2, or 1+5*66, or 331.
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PROGRAM
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In Iverson's J language: Let cfn be the formula above. Then the first 20 rows and columns of T are: T =: cfn / ~ i. 20 where i. 20 gives the first 20 nonnegative integers, the tilde (~) is Curry's W, or duplicate, which makes i. 20 also the second argument, and the slash (/) gives the outer product with cfn of each item of the first argument with each item of the second argument, in this case the first 20 rows and columns of T.
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CROSSREFS
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Anti-diagonal sums are in A101338.
The following entries are all for centered figurate numbers for various m-gons. They are A000124(1), A002061(2), A005448(3) A001844(4), A005891(5), A003215(6), A069099(7), A016754(8), A060544(9), A062789(10), A069125(11), A003154(12), A069126(13), A069127(14), A069128(15), A069129(16), A069130(17), A069131(18), A069132(19) and A069133(20). Column 2 of T is A016777.
Cf. A001263.
Sequence in context: A034367 A058717 A034371 this_sequence A091186 A138155 A055587
Adjacent sequences: A101318 A101319 A101320 this_sequence A101322 A101323 A101324
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KEYWORD
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easy,nonn,tabl,uned
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AUTHOR
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Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004
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