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%I #56 Mar 21 2024 21:06:02
%S 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,
%T 104,192,346,14,24,42,73,129,233,425,771,15,29,53,95,168,297,530,955,
%U 1726,19,34,63,116,211,379,676,1206,2161,3887,16,35,69,132,248,459,838
%N Zorach additive triangle, read by rows.
%C Each number is sum of west and northwest numbers; each number appears at most once in table.
%C Conjectured to form a permutation of the positive integers.
%C Number in column 1 is least so that there are no repeats in the row.
%C Inverse of sequence A035358 considered as a permutation of the positive integers. - _Howard A. Landman_, Sep 25 2001
%C 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
%C This is the sequence generated by applying Jackson's difference fan transformation to A035313. - _David W. Wilson_, Feb 26 2012
%C Using data from the first 300 rows, it appears that the least number not yet used is not greater than but asymptotically equal to twice the row number. (The least unused number in rows 1 through 299 is 592.) - _M. F. Hasler_, May 09 2013
%C Row n is the binomial transform of the first n terms of A035311, reversed. - _Andrey Zabolotskiy_, Feb 09 2017
%H Reinhard Zumkeller, <a href="/A035312/b035312.txt">Rows n=0..150 of triangle, flattened</a>
%H E. Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2013-May/011131.html">Three triangles</a>, SeqFan list, May 8, 2013
%H Chris Zheng, Jeffrey Zheng, <a href="https://doi.org/10.1007/978-981-13-2282-2_4">Triangular Numbers and Their Inherent Properties</a>, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.
%H A. C. Zorach, <a href="http://www.cazort.net/static/triangle.php">Additive triangle</a>
%H Reinhard Zumkeller, <a href="/A035312/a035312_2.hs.txt">Haskell programs for sequences in connection with Zorach additive triangle</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e Triangle begins:
%e 1;
%e 2, 3;
%e 4, 6, 9;
%e 7, 11, 17, 26;
%e 5, 12, 23, 40, 66;
%e 8, 13, 25, 48, 88, 154;
%e 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...
%t (* 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}]] (* _Jean-François Alcover_, Dec 02 2011 *)
%o See link for Haskell program.
%o (PARI) {u=a=[l=1]; for(n=1,20,print(a); a[1]==l && while(setsearch(u,l++),); s=l; while(setintersect(u,t=vector(1+n,i,if(i<2,t=s,t+=a[i-1]))),s++); u=setunion(u,a=t))} \\ _M. F. Hasler_, May 09 2013
%Y Cf. A035311 (left edge), A035313 (right edge), A189713 (central), A189714 (row sums), A072038, A072039.
%K nonn,tabl,easy,nice
%O 0,2
%A _Alex Zorach_
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