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Search: id:A144112
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| A144112 |
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Weight array W={w(i,j)} of the natural number array A000027. |
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+0
13
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| 1, 1, 2, 2, 1, 3, 3, 1, 1, 4, 4, 1, 1, 1, 5, 5, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 1, 1, 8, 8, 1, 1, 1, 1, 1, 1, 1, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 10, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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In general, let w(i,j) be the weight of the unit square labeled by its
northeast vertex (i,j) and for each (m,n), define
S(m,n)=SUM{SUM{w(i,j), i=1,2,...,m, j=1,2,...,n}.
Then S(m,n) is the weight of the rectangle [0,m]x[0,n]. We call W the weight
array of S and we call S the accumulation array of W. For the case at hand, S is
the array of natural numbers having the following antidiagonals:
(1), then (2,3), then (4,5,6), then (7,8,9,10) and so on.
Contribution from Clark Kimberling (ck6(AT)evansville.edu), Sep 14 2008: (Start)
In general, the weight array W of an arbitrary rectangular array R={R(i,j)}, given by i>=1, j>=1, is defined in two steps:
(1) put R(i,j)=0 if i=0 or j=0;
(2) then w(m,n)=R(m,n)+R(m-1,n-1)-R(m,n-1)-R(m-1,n) for m>=1, n>=1. (End)
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FORMULA
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row 1: 1 followed by A000027
row n: n followed by A000012, for n>1.
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EXAMPLE
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S(2,4)=1+1+2+3+2+1+1+1=12.
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CROSSREFS
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Cf. A000012, A000027.
Sequence in context: A145141 A103360 A104469 this_sequence A104660 A093613 A118816
Adjacent sequences: A144109 A144110 A144111 this_sequence A144113 A144114 A144115
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), Sep 11 2008
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