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A035312 Zorach additive triangle, read by rows.

%I A035312 #59 Dec 23 2024 18:12:57
%S A035312 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 A035312 104,192,346,14,24,42,73,129,233,425,771,15,29,53,95,168,297,530,955,
%U A035312 1726,19,34,63,116,211,379,676,1206,2161,3887,16,35,69,132,248,459,838
%N A035312 Zorach additive triangle, read by rows.
%C A035312 Each number is sum of west and northwest numbers; each number appears at most once in table.
%C A035312 Conjectured to form a permutation of the positive integers.
%C A035312 Number in column 1 is least so that there are no repeats in the row.
%C A035312 Inverse of sequence A035358 considered as a permutation of the positive integers. - _Howard A. Landman_, Sep 25 2001
%C A035312 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 A035312 This is the sequence generated by applying Jackson's difference fan transformation to A035313. - _David W. Wilson_, Feb 26 2012
%C A035312 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 A035312 Row n is the binomial transform of the first n terms of A035311, reversed. - _Andrey Zabolotskiy_, Feb 09 2017
%H A035312 Reinhard Zumkeller, <a href="/A035312/b035312.txt">Rows n=0..150 of triangle, flattened</a>
%H A035312 E. Angelini, <a href="https://web.archive.org/web/20220117113613/http://list.seqfan.eu/pipermail/seqfan/2013-May/011131.html">Three triangles</a>, SeqFan list, May 8, 2013
%H A035312 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 A035312 A. C. Zorach, <a href="http://www.cazort.net/static/triangle.php">Additive triangle</a>
%H A035312 Reinhard Zumkeller, <a href="/A035312/a035312_2.hs.txt">Haskell programs for sequences in connection with Zorach additive triangle</a>
%H A035312 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A035312 Triangle begins:
%e A035312   1;
%e A035312   2,  3;
%e A035312   4,  6,  9;
%e A035312   7, 11, 17, 26;
%e A035312   5, 12, 23, 40, 66;
%e A035312   8, 13, 25, 48, 88, 154;
%e A035312 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 A035312 (* 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 A035312 See link for Haskell program.
%o A035312 (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 A035312 Cf. A035311 (left edge), A035313 (right edge), A189713 (central), A189714 (row sums), A072038, A072039.
%K A035312 nonn,tabl,easy,nice
%O A035312 0,2
%A A035312 _Alex Zorach_