OFFSET
0,2
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, 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
This is the sequence generated by applying Jackson's difference fan transformation to A035313. - David W. Wilson, Feb 26 2012
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
Row n is the binomial transform of the first n terms of A035311, reversed. - Andrey Zabolotskiy, Feb 09 2017
LINKS
Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
E. Angelini, Three triangles, SeqFan list, May 8, 2013
Chris Zheng, Jeffrey Zheng, Triangular Numbers and Their Inherent Properties, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.
A. C. Zorach, Additive triangle
Reinhard Zumkeller, Haskell programs for sequences in connection with Zorach additive triangle
EXAMPLE
Triangle begins:
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...
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}]] (* Jean-François Alcover, Dec 02 2011 *)
PROG
See link for Haskell program.
(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
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved