Alex Zorach - cp's OEIS frontend

User: Alex Zorach

Alex Zorach has authored 3 sequences.

A035311 First column of the Zorach additive triangle A035312.

1, 2, 4, 7, 5, 8, 10, 14, 15, 19, 16, 20, 21, 22, 27, 30, 28, 33, 32, 38, 37, 39, 44, 45, 51, 46, 47, 54, 55, 52, 59, 62, 60, 64, 50, 67, 72, 78, 74, 68, 81, 82, 80, 85, 79, 87, 86, 91, 90, 94, 102, 98, 103, 99, 105, 100, 108, 112, 110, 113, 127, 128, 125, 133, 118, 123

Offset: 0

Alex Zorach

One can see that (least unused number in rows 1 through n of A035312) ~ A035311(n) ~ 2n. (Asymptotic equality, and the first quantity does not exceed either of the two others.) - M. F. Hasler, May 09 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035312, A035313, A189713, A035358.

Haskell
```
-- See link for Haskell program.
```

(* Assuming n <= t(n,1) <= 3n *) 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, 3 n], Flatten[ Table[t[m, k], {m, 1, n - 1}, {k, 1, m}]]], uniqueQ[#, n] & , 1][[1]]; Table[t[n, 1], {n, 1, 66}](* Jean-François Alcover, Dec 02 2011 *)

More terms from Christian G. Bower

A035313 (Largest) diagonal of the Zorach additive triangle A035312.

Original entry on oeis.org

1, 3, 9, 26, 66, 154, 346, 771, 1726, 3887, 8768, 19700, 43890, 96717, 210665, 453893, 968903, 2053260, 4328489, 9093971, 19068611, 39943689, 83628399, 175018523, 366081209, 765102907, 1597315656, 3330380593, 6933810145

Offset: 0

Alex Zorach

From Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 22 2007: (Start)
Starting with 1, smallest sequence for which:
all its terms a1(n).............................. 1,3,9,26,66
all terms of first differences a2(n)=a1(n+1)-a1(n) 2,6,17,40
all terms of second differences a3(n)=a2(n+1)-a2(n) 4,11,23
...
all terms of (1+i)th differences ai(n)=ai-1(n+1)-ai-1(n)
are different for any n and any i (End)
Which is to say, this sequence is the lexicographically earliest sequence of positive integers such that the sequence itself and its n-th differences for n >= 1 are pairwise disjoint. - David W. Wilson, Feb 26 2012
Conjecturally, every positive integer occurs in the sequence or one of its n-th differences, which would imply that the sequence and its n-th differences partition the positive integers. - David W. Wilson, Feb 26 2012
Conjecture: lim(n->infinity, a(n+1)/a(n)) = 2. - David W. Wilson, Feb 26 2012
Note that the n-th differences yield the n-th subdiagonals (parallels to the right edge) in the triangle A035312. Therefore Lallouet's statement and Wilson's 1st comment above are just rephrasing the definition of that triangle.  - M. F. Hasler, May 09 2013
Binomial transform of A035311. Hence, from the observed asymptotic equality A035311(n) ~ 2*n, a stronger statement than the one given above follows: a(n) ~ n*2^n. - Andrey Zabolotskiy, Feb 08 2017

			Start with 1; 2 is the next, then add 1+2 to get 3, then 4 is next, then 4+2=6 and 6+3 is 9, then 5 is not next because 5+4=9 and 9 was already used, so 7 is next...which ultimately generates 26 in the final column...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. C. Zorach, Additive triangle
A. C. Zorach, Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035311, A035312, A189713, A035358.

Haskell
```
-- See link for Haskell program.
```

(* Assuming n <= t(n,1) <= 3n *) rows = 29; 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, 3 n], Flatten[ Table[t[m, k], {m, 1, n-1}, {k, 1, m}]]], uniqueQ[#, n] &, 1][[1]]; Last /@ Table[t[n, k], {n, 1, rows}, {k, 1, n}] (* Jean-François Alcover, Jun 05 2012 *)

More terms from Christian G. Bower and Dean Hickerson

A035312 Zorach additive triangle, read by rows.

Original entry on oeis.org

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

Offset: 0

Alex Zorach

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

			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...

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
Index entries for sequences that are permutations of the natural numbers

Cf. A035311 (left edge), A035313 (right edge), A189713 (central), A189714 (row sums), A072038, A072039.

(* 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 *)

{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

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User: Alex Zorach

A035311 First column of the Zorach additive triangle A035312.

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A035313 (Largest) diagonal of the Zorach additive triangle A035312.

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A035312 Zorach additive triangle, read by rows.

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