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A035312
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Zorach additive triangle, read by rows.
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11
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1, 2, 3, 4, 6, 9, 7, 11, 17, 26, 5, 12, 23, 40, 66, 8, 13, 25, 48, 88, 154, 10, 18, 31, 56, 104, 192, 346, 14, 24, 42, 73, 129, 233, 425, 771, 15, 29, 53, 95, 168, 297, 530, 955, 1726, 19, 34, 63, 116, 211, 379, 676, 1206, 2161, 3887, 16, 35, 69, 132, 248, 459, 838
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Each number is sum of west and northwest numbers; each number appears at most once in table.
Conjectured to form a permutation of the positive integers.
Number in column 1 is least so that there are no repeats in the row.
Inverse of sequence A035358 considered as a permutation of the positive integers. - Howard A. Landman (howard(AT)polyamory.org), Sep 25 2001
The following four statements are equivalent, (all n): (i) A035358(n)>0, (ii) A072038(n)>0, (iii) A072039(n)>0, (iv) the flattened triangle is a permutation of the natural numbers; in this case the inverse is A035358 and A035358(n)=A000217(A072039(n)-1)+A072038(n) . [Reinhard Zumkeller, Apr 30 2011]
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LINKS
| A. C. Zorach, Additive triangle
Index entries for sequences that are permutations of the natural numbers
Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Reinhard Zumkeller, Haskell programs for sequences in connection with Zorach additive triangle
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EXAMPLE
| 1; 2,3; 4,6,9; 7,11,17,26; 5,12,23,40,66; 8,13,25,48,88,154; ...
E.g. 1 is the first number, 2 is the next, then add 1+2 to get 3, then 4 is next, then 4+2=6, 6+3=9, then 5 is not next because 5+4=9 and 9 was already used, so 7 is next...
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MATHEMATICA
| (* Assuming n <= t(n, 1) <= 3n *) rows = 11; uniqueQ[t1_, n_] := (t[n, 1] = t1; Do[t[n, k] = t[n, k-1] + t[n-1, k-1], {k, 2, n}]; n*(n+1)/2 == Length[ Union[ Flatten[ Table[ t[m, k], {m, 1, n}, {k, 1, m}]]]]); t[n_ , 1] := t[n, 1] = Select[ Complement[ Range[n, 3n], Flatten[ Table[ t[m, k], {m, 1, n-1}, {k, 1, m}]]], uniqueQ[#, n]& , 1][[1]]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* From Jean-François Alcover, Dec 02 2011 *)
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PROG
| See link for Haskell program.
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CROSSREFS
| Cf. A035311 (left edge), A035313 (right edge), A189713 (central), A189714 (row sums).
Sequence in context: A141396 A159849 A098168 * A056230 A119919 A036561
Adjacent sequences: A035309 A035310 A035311 * A035313 A035314 A035315
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| Alexander C. Zorach (cazort(AT)udel.edu)
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