A035312 -id:A035312 - cp's OEIS frontend

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.

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

A235538 Earliest infinite sequence of natural numbers such that the members of this sequence as well as the absolute values of the members of the k-th differences of this sequence, for all k>0, are all distinct.

Original entry on oeis.org

1, 3, 9, 26, 5, 13, 31, 15, 27, 81, 22, 45, 92, 20, 50, 145, 46, 89, 32, 71, 151, 40, 75, 163, 73, 124, 60, 126, 244, 97, 219, 63, 132, 306, 68, 144, 297, 79, 166, 354, 83, 187, 394, 94, 203, 419, 108, 220, 460, 127, 260, 110, 247, 513, 161, 340, 117, 252

Offset: 1

Paul Tek, Jan 12 2014

			For n=1:
- 1 is admissible; hence a(1)=1.
For n=2:
- 1 is not admissible (as it already appears in the sequence),
- 2 is not admissible (as a(1) would appear in the first differences),
- 3 is admissible; hence a(2)=3.
For n=3:
- 1 is not admissible (as it already appears in the sequence),
- 2 is not admissible (as it already appears in the first differences),
- 3 is not admissible (as it already appears in the sequence),
- 4 is not admissible (as a(1) would appear in the first differences),
- 5 is not admissible (as 2 would appear twice in the first differences),
- 6 is not admissible (as a(2) would appear in the first differences),
- 7 is not admissible (as 2 would appear in the first and second differences),
- 8 is not admissible (as a(2) would appear in the second differences),
- 9 is admissible; hence a(3)=9.

Paul Tek, Table of n, a(n) for n = 1..1000
Paul Tek, PERL program for this sequence

Cf. A005228, A005282, A035312, A235539, A327460, A327887.

a[1] = 1; diffs0 = {1} (* flattened array of successive differences *);
a[n_] := a[n] = Module[{}, aa = Array[a, n-1]; m0 = 1; While[ MemberQ[ diffs0, m0], m0++]; For[m = m0, True, m++, am = Append[aa, m]; td = Table[Differences[am, k], {k, 0, n-1}]; diffs = Abs[Flatten[td]]; If[ Length[diffs] == Length[Union[diffs]], diffs0 = diffs//Sort; Return[m]]] ];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 31 2018 *)

Perl
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See Link section.
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Added "infinite" to definition. - N. J. A. Sloane, Oct 05 2019

A375761 Infinite triangle T(n, k), n > 0, k = 1..n, read and filled in the greedy way by rows with distinct positive integers such that for any n > 1, k = 2..n, T(n, k) = T(n-1, k-1) * T(n, k-1).

Original entry on oeis.org

2, 3, 6, 4, 12, 72, 5, 20, 240, 17280, 7, 35, 700, 168000, 2903040000, 8, 56, 1960, 1372000, 230496000000, 669139107840000000000, 10, 80, 4480, 8780800, 12047257600000, 2776844687769600000000000, 1858095376984393503473664000000000000000000000

Offset: 1

Rémy Sigrist, Aug 27 2024

This sequence is a variant of A035312, based on products instead of sums.
The value 1 cannot appear (as it would imply a duplicate term).
We can always start a new row with a prime number that does not divide prior terms, so the sequence is well defined.
Will every integer > 1 appear in the sequence?

			Triangle T(n, k) begins:
               2
           3      6
         4   12      72
       5  20    240      17280
     7  35  700    168000    2903040000
    8 56 1960 1372000 230496000000 669139107840000000000

Rémy Sigrist, PARI program

Cf. A035312, A064319, A066117, A375846 (right border), A375857 (left border).

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

A136563 Leftmost column of triangle A136561.

Original entry on oeis.org

1, 2, 4, -5, 13, -30, 75, -200, 524, -1299, 3038, -6762, 14482, -30162, 61691, -124998, 252744, -512805, 1047661, -2158235, 4481801, -9368303, 19674518, -41437744, 87399483, -184412912, 388975790, -819684340, 1724741941, -3621720214, 7585569484, -15839701483

Offset: 1

Leroy Quet, Jan 06 2008

sign

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 leftmost column of the Zorach additive triangle is A035311.

Cf. A035311, A136561, A136562.

More terms from Andrey Zabolotskiy, May 29 2017

A185290 Lexicographically least permutation of the integers in a triangle satisfying T(n,k) + T(n+1,k) <= T(n+1,k+1).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 10, 16, 25, 7, 12, 22, 38, 63, 8, 15, 27, 49, 87, 150, 11, 19, 34, 61, 110, 197, 347, 13, 24, 43, 77, 138, 248, 445, 792, 14, 28, 52, 95, 172, 310, 558, 1003, 1795, 17, 31, 59, 111, 206, 378, 688, 1246, 2249, 4044, 18, 35, 66, 125, 236, 442, 820, 1508, 2754, 5003, 9047, 20, 39, 74, 140, 265, 501, 943, 1763

Offset: 1

Paul Tek, Apr 23 2013

Apparently, T(n,k) ~ n*2^k as n grows.

			The first rows are:
[1]
[2,3]
[4,6,9]
[5,10,16,25]
For T(1,1), we choose the first integer not yet used: 1.
For T(2,1), we choose the first integer not yet used: 2.
For T(2,2), we choose the first integer >= T(1,1)+T(2,1) not yet used: 3.
For T(3,1), we choose the first integer not yet used: 4.
For T(3,2), we choose the first integer >= T(2,1)+T(3,1) not yet used: 6.
For T(3,3), we choose the first integer >= T(2,2)+T(3,2) not yet used: 9.
etc.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A035312.

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

A275705 A variation of the Zorach additive triangle, read by rows.

Original entry on oeis.org

1, -2, -1, -3, -5, -6, -4, -7, -12, -18, -9, -13, -20, -32, -50, 11, 2, -11, -31, -63, -113, 3, 14, 16, 5, -26, -89, -202, 6, 9, 23, 39, 44, 18, -71, -273, 4, 10, 19, 42, 81, 125, 143, 72, -201, 8, 12, 22, 41, 83, 164, 289, 432, 504, 303, 7, 15, 27, 49, 90

Offset: 1

Max Barrentine, Aug 06 2016

This is a variation of the Zorach additive triangle (A035312), with negative numbers included. Each term is the sum of the terms to its immediate west and northwest, and each member of the first column is chosen such that its absolute value is minimal and no term following it in its own row occurs earlier in the triangle. In the case where m and -m both satisfy these criteria for T(n,1) = |m|, choose the term that minimizes the absolute value of T(n,2).

Is this a permutation of the nonzero integers?

Max Barrentine, Rows n = 1..50 of triangle, flattened

Cf. A035312.

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A235538 Earliest infinite sequence of natural numbers such that the members of this sequence as well as the absolute values of the members of the k-th differences of this sequence, for all k>0, are all distinct.

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A375761 Infinite triangle T(n, k), n > 0, k = 1..n, read and filled in the greedy way by rows with distinct positive integers such that for any n > 1, k = 2..n, T(n, k) = T(n-1, k-1) * T(n, k-1).

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A136563 Leftmost column of triangle A136561.

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A185290 Lexicographically least permutation of the integers in a triangle satisfying T(n,k) + T(n+1,k) <= T(n+1,k+1).

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

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Extensions

A275705 A variation of the Zorach additive triangle, read by rows.

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