

A035312


Zorach additive triangle, read by rows.


17



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
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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.
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


LINKS



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, k1] + t[n1, k1], {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, n1}, {k, 1, m}]]], uniqueQ[#, n]& , 1][[1]]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]] (* JeanFrançois 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[i1]))), s++); u=setunion(u, a=t))} \\ M. F. Hasler, May 09 2013


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



