A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10

Offset

0,6

Comments

A formal extension of the figurative numbers A139600 to negative n.

Links

Table of n, a(n) for n=0..65.

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Example

[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Maple

A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);

Prog

(Python 3)
def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

Crossrefs

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).

Columns include A008585, A016933, A017329.

Cf. A139600.

Sequence in context: A316827 A286628 A180243 * A330944 A064918 A323076

Adjacent sequences: A326725 A326726 A326727 * A326729 A326730 A326731

Keyword

sign,tabl,easy

Author

Peter Luschny, Aug 04 2019

Status

approved