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

1,2

Comments

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

Links

Table of n, a(n) for n=1..50.

Example

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.

Prog

(Python)
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

Crossrefs

Cf. A035313, A136561, A136563.

Sequence in context: A294480 A195972 A274435 * A305342 A197274 A316234

Adjacent sequences: A136559 A136560 A136561 * A136563 A136564 A136565

Keyword

nonn

Author

Leroy Quet, Jan 06 2008

Extensions

More terms from Andrey Zabolotskiy, May 29 2017

Status

approved