A035313 -id:A035313 - cp's OEIS frontend

A035312 Zorach additive triangle, read by rows.

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

A327460 Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, all the k(k+1)/2 numbers in the triangle of differences of the first k terms are distinct.

Original entry on oeis.org

1, 3, 9, 5, 12, 10, 23, 8, 22, 17, 42, 16, 43, 20, 38, 26, 45, 32, 65, 28, 64, 39, 76, 34, 81, 48, 98, 40, 92, 54, 109, 60, 116, 51, 114, 58, 117, 70, 136, 67, 135, 71, 145, 72, 147, 69, 146, 80, 164, 87, 166, 82, 170, 108, 198, 101

Offset: 1

N. J. A. Sloane, Sep 25 2019

This is an infinite version of A327762. The first 55 terms are the same as in A327762.
Inspired by A327743.
The usual topological arguments show that there IS a sequence satisfying the definition. So far, the terms of A327460 lie on two roughly straight lines, of slopes about 1.75 and 3.5: see A328069, A328070. - N. J. A. Sloane, Oct 07 2019
If only the first differences are constrained, one gets the classical Mian-Chowla sequence A005282. - M. F. Hasler, Oct 09 2019. See also another classic, A005228, and A328190. - N. J. A. Sloane, Nov 01 2019

			The difference triangle of the first k=8 terms of the sequence is
     1,    3,    9,   5,  12,  10,  23, 8, ...
     2,    6,   -4,   7,  -2,  13, -15, ...
     4,  -10,   11,  -9,  15, -28, ...
   -14,   21,  -20,  24, -43, ...
    35,  -41,   44, -67, ...
   -76,   85, -111, ...
   161, -196, ...
  -357, ...
All 8*9/2 = 36 numbers are distinct.

Peter Kagey, Table of n, a(n) for n = 1..4000

Cf. A005228, A005282, A035313, A327743, A327762, A328190.
See also A327458 (differences), A328066 (sorted), A328067, A328068 (complement), A328069 and A328070 (bisections), A328071; A235538 (absolute differences distinct).
The inverse binomial transform is A327459.

A035311 First column of the Zorach additive triangle A035312.

Original entry on oeis.org

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

A035358 k such that A035312(k-1) = n or 0 if there is none.

Original entry on oeis.org

1, 2, 3, 4, 11, 5, 7, 16, 6, 22, 8, 12, 17, 29, 37, 56, 9, 23, 46, 67, 79, 92, 13, 30, 18, 10, 106, 137, 38, 121, 24, 172, 154, 47, 57, 68, 211, 191, 232, 14, 80, 31, 93, 254, 277, 326, 352, 19, 107, 596, 301, 436, 39, 379, 407, 25, 122, 138, 466, 529, 155, 497, 48

Offset: 1

Christian G. Bower, Nov 15 1998

nonn

At least up through the first 19 terms (ending at 46), this appears to be identical to the inverse of sequence A035312 considered as a permutation of the positive integers. - Howard A. Landman, Sep 23 2001

This sequence is *by definition* the inverse of A035312 (upon shifting its offset), provided that A035312 indeed is surjective on the positive integers. - M. F. Hasler, May 09 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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, A035312, A035313.

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

a(n) = A000217(A072039(n) - 1) + A072038(n) if the sequence is in fact the inverse permutation of the flattened Zorach additive triangle. - Reinhard Zumkeller, Apr 30 2011

A072038 Column of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 3, 4, 1, 1, 2, 1, 3, 1, 1, 2, 2, 2, 1, 1, 1, 4, 2, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 3, 1, 1, 4, 2, 2, 1, 1, 2, 1, 3, 1, 2, 5, 1, 1, 3, 2, 3, 1, 4, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 5, 2, 1, 1, 3, 2, 1, 4, 2, 2, 1, 1, 1, 2, 1, 1, 5, 1

Offset: 1

Henry Bottomley, Jun 07 2002

nonn

			a(8)=1 since 8 appears in the first (i.e. left hand) column.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

If a(n)=1 then n is in A035311, if a(n)=A072039(n) then n is in A035313. Cf. A035312, A035358, A072039.

Haskell
```
-- See link.
```

If a(n) > 0: n = A035312(A072039(n)-1, a(n)-1).

A072039 Row of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 6, 3, 7, 4, 5, 6, 8, 9, 11, 4, 7, 10, 12, 13, 14, 5, 8, 6, 4, 15, 17, 9, 16, 7, 19, 18, 10, 11, 12, 21, 20, 22, 5, 13, 8, 14, 23, 24, 26, 27, 6, 15, 35, 25, 30, 9, 28, 29, 7, 16, 17, 31, 33, 18, 32, 10, 34, 19, 5, 36, 40, 11, 20, 12, 37, 8, 39, 21, 22, 13, 38, 45, 43

Offset: 1

Henry Bottomley, Jun 07 2002

nonn

			a(8)=6 since 8 appears in the sixth row.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035312, A035311, A035313, A035358, A072038.

Haskell
```
-- See link.
```

If a(n) > 0: n = A035312(a(n)-1, A072038(n)-1).

A189713 Central terms of Zorach additive triangle (cf. A035312).

Original entry on oeis.org

1, 6, 23, 56, 168, 459, 1080, 2400, 5461, 12678, 28728, 63336, 138717, 303346, 663179, 1438098, 3062730, 6450121, 13583957, 28627773, 59954459, 124413466, 258510987, 543867101, 1152248068, 2428235533, 5069886692, 10511339217, 21696371682, 44710463120

Offset: 0

Reinhard Zumkeller, Apr 30 2011

nonn

a(n) = A035312(2*n,n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
A. C. Zorach, Additive triangle
Haskell programs for sequences in connection with Zorach additive triangle

Cf. A035311, A035313.

Haskell
```
-- See link.
```

A136562 Consider the triangle A136561: the n-th diagonal (from the right) is the sequence of (signed) differences between pairs of consecutive terms in the (n-1)th diagonal. The rightmost diagonal (A136562) is defined: A136562(1)=1; A136562(n) is the smallest integer > A136562(n-1) such that any (signed) integer occurs at most once in the triangle A136561.

Original entry on oeis.org

1, 3, 9, 14, 26, 36, 63, 74, 103, 118, 149, 169, 210, 233, 280, 302, 357, 392, 464, 489, 553, 591, 673, 713, 796, 844, 941, 987, 1083, 1134, 1238, 1292, 1398, 1463, 1596, 1652, 1769, 1840, 1980, 2046, 2172, 2250, 2416, 2492, 2565, 2715, 2836, 3051, 3130, 3298

Offset: 1

Leroy Quet, Jan 06 2008

nonn

Requiring that the absolute values of the differences in the difference triangle only occur at most once each leads to the Zorach additive triangle. (See A035312.) The rightmost diagonal of the Zorach additive triangle is A035313.

It appears that a(n) is proportional to n^2. - Andrey Zabolotskiy, May 29 2017

			The triangle begins:
1,
2,3,
4,6,9,
-5,-1,5,14,
13,8,7,12,26,
-30,-17,-9,-2,10,36.
Example:
Considering the rightmost value of the 4th row: Writing a 10 here instead, the first 4 rows of the triangle become:
1
2,3
4,6,9
-9,-5,1,10
But 1 already occurs earlier in the triangle. So 10 is not the rightmost element of row 4.
Checking 11,12,13,14; 14 is the smallest value that can be the rightmost element of row 4 and not have any elements of row 4 occur earlier in the triangle. So A136562(4) = 13.

Cf. A035313, A136561, A136563.

a, t = [1], [1]
for n in range(1, 100):
    d = a[-1]
    while True:
        d += 1
        row = [d]
        for j in range(n):
            row.append(row[-1]-t[-j-1])
            if row[-1] in t:
                break
        else:
            a.append(d)
            t += reversed(row)
            break
print(a)
# t contains the triangle
# [t[n*(n-1)/2] for n in range(1, 100)] gives leftmost column
# Andrey Zabolotskiy, May 29 2017

More terms from Andrey Zabolotskiy, May 29 2017

A375846 Right border of triangle A375761.

Original entry on oeis.org

2, 6, 72, 17280, 2903040000, 669139107840000000000, 1858095376984393503473664000000000000000000000, 17605663024171636760609250204915381895780997959633600512000000000000000000000000000000000000000000

Offset: 1

Rémy Sigrist, Aug 31 2024

nonn

The terms of this sequence, the ratios of consecutive terms, the ratios of ratios of consecutive terms, etc. are all distinct integers.

			a(3) = A375761(3, 3) = 72.

Rémy Sigrist, PARI program

Cf. A035313, A375761, A375857.

PARI
```
\\ See Links section.
```

a(n) = A375761(n, n).

a(n) = Product_{k = 1..n} A375857(k)^binomial(n-1, k-1).

A226239 Minimum m such that there exists an n-row subtractive triangle with distinct integers in 1..m.

Original entry on oeis.org

1, 3, 6, 10, 15, 22, 33, 44, 59, 76, 101, 125, 158

Offset: 1

Yi Yang, Jun 01 2013

In an n-row subtractive triangle, there are n-i+1 integers in the i-th row. The integers in the first row are arbitrary. From the next row, the integers are the absolute difference between adjacent integers in the previous row.

			a(6)=22 because there is a 6-row subtractive triangle with distinct integers in [1..22] as follows:
1:  6 20 22  3 21 13
2: 14  2 19 18  8
3: 12 17  1 10
4:  5 16  9
5: 11  7
6:  4
However, there is no such triangle with distinct integers in [1..21].

Chyanog, A Chinese web page where the problem was posed.
International Mathematical Olympiad, Problem 3 of IMO 2018.
Denis Cazor, Algorithme en Français
Denis Cazor, Algorithm in English

Cf. A035312, A035313.

a(12) from Yi Yang, Mar 04 2015

a(13) from Denis Cazor, Aug 01 2022

cp's OEIS Frontend

A035312 Zorach additive triangle, read by rows.

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A327460 Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, all the k(k+1)/2 numbers in the triangle of differences of the first k terms are distinct.

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A035311 First column of the Zorach additive triangle A035312.

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A035358 k such that A035312(k-1) = n or 0 if there is none.

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A072038 Column of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

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A072039 Row of Zorach additive triangle in which n appears (or 0 in the unlikely event that it does not).

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A189713 Central terms of Zorach additive triangle (cf. A035312).

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A375846 Right border of triangle A375761.