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A125585
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Array of constant-spaced integers read by antidiagonals.
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1
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1, 1, 2, 2, 3, 3, 1, 4, 5, 4, 2, 4, 6, 7, 5, 3, 5, 7, 8, 9, 6, 1, 6, 8, 10, 10, 11, 7, 2, 5, 9, 11, 13, 12, 13, 8, 3, 6, 9, 12, 14, 16, 14, 15, 9, 4, 7, 10, 13, 15, 17, 19, 16, 17, 10, 1, 8, 11, 14, 17, 18, 20, 22, 18, 19, 11, 2, 6, 12, 15, 18, 21, 21, 23, 25, 20, 21, 12
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refs;
listen;
history;
text;
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OFFSET
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1,3
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COMMENTS
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Iteratively taking sums of the values in each row starting with 1 produces the "figurate" numbers. For instance: 1, 1 + 2 = 3, 1 + 2 + 3 = 6 (the Triangular numbers -- A000217) 1, 1 + 3 = 4, 1 + 3 + 5 = 9 (the Square numbers -- A000290) 1, 1 + 4 = 5, 1 + 4 + 7 = 10 (the Pentagonal numbers -- A000326) etc. Iterative sums of the rows in between produce sequences related to the figurate numbers: 2, 2+4=6, 2+4+6=10 (oblong, or pronic, or heteromecic numbers -- A002378) 2, 2+5=7, 2+5+8=15 (second pentagonal numbers -- A005449) 3, 3+6=9, 3+6+9=18 (triangular matchstick numbers --A045943) etc. Iterative products produce the n-Factorial numbers: 1, 1*3=3, 1*3*5=15 (Double factorial numbers: (2n-1)!! -- A001147 2, 2*4=8, 2*4*6=48 (Double factorial numbers: (2n)!! -- A000165) 1, 1*4=4, 1*4*7=28, (Triple factorial numbers (3*n-2)!!! -- A007559) etc.
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LINKS
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EXAMPLE
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The array begins:
1, 2, 3, 4, 5, 6, ...
1, 3, 5, 7, 9, 11, ...
2, 4, 6, 8, 10, 12, ...
1, 4, 7, 10, 13, 16, ...
2, 5, 8, 11, 14, 17, ...
3, 6, 9, 12, 15, 18, ...
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MAPLE
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A:= proc(n, k) local m;
m:= floor((sqrt(8*n-7)-1)/2);
n + (m+1)*(k-1-m/2)
end:
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MATHEMATICA
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imax = 5;
A = Table[k, {i, 1, imax}, {j, 1, i}, {k, j, j + i*imax*(imax+1)/2 - 1, i} ] // Flatten[#, 1]&;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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