A089595 Table T(n,k), n>=0 and k>=0: Stern's diatomic array read by antidiagonals (version 5).

1, 0, 1, -1, 1, 1, -3, 0, 2, 1, -2, -1, 1, 3, 1, -7, -1, 1, 2, 4, 1, -5, -4, 0, 3, 3, 5, 1, -8, -3, -1, 1, 5, 4, 6, 1, -3, -5, -1, 2, 2, 7, 5, 7, 1, -13, -2, -2, 1, 5, 3, 9, 6, 8, 1, -10, -9, -1, 1, 3, 8, 4, 11, 7, 9, 1, -17, -7, -5, 0, 4, 5, 11, 5, 13, 8, 10, 1, -7, -12, -4, -1, 1, 7, 7, 14, 6, 15, 9, 11, 1, -18, -5, -7, -1, 3, 2, 10, 9, 17, 7

Offset

0,7

Links

Table of n, a(n) for n=0..100.

Formula

Each row is obtained by copying the previous row but interpolating the sum of pairs of adjacent terms.

T(n, 2*k) = T(n-1, k) = T(n, k) - A002487(k).

T(n, 2*k+1) = T(n, 2*k) + T(n, 2*k+2); T(0, 0)=1, T(0, 1)=0.

The k-th column is an arithmetic progression with: T(n, k) = T(0, k) + n*A002487(k).

From William P. Orrick, May 14 2026: (Start)

Negation of row n = 0 is A156140.

Negation of row n = 1 is A321154.

T(n, 2^n - j) = -T(n, 2^n + j) = A002487(j) for n >= 0, 0 <= j <= 2^n.

T(n, k) = 1 if k = 0; T(n, k) = T(r, c) * (n - r + 2) - T(r+1, c), where r = floor(log_2(k)), c = n - 2^(r - 1), if k > 0. (End)

Example

Table array begins
  row n=0: 1, 0, -1, -3, -2, -7, -5, -8, -3, -13, -10, -17, -7, -18, -11, ...
  row n=1: 1, 1, 0, -1, -1, -4, -3, -5, -2, -9, -7, -12, -5, -13, ...
  row n=2: 1, 2, 1, 1, 0, -1, -1, -2, -1, -5, -4, -7, -3, ...
  row n=3: 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, ...
  row n=4: 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, ...
  ...
From William P. Orrick, May 14 2026: (Start)
Negations of rows may be read off from the denominators of region labels in horizontal transections of the topograph as presented below. In every transection the sequence of numerators is A002487.
       |_____________________________________________________________________________
       |
       A                              1/2
       |_________________________________________________________________
       |                                                                 |
       B                      1/1                                        |
       |_____________________________________                     _______|_______
       |                                     |                   |               |
       C              1/0                    |                   |      2/3      |
0/(-1) |__________________              _____|______          ___|___         ___|___
       |                  |            |            |        |       |       |       |
       D       1/(-1)     |            |     2/1    |        |  3/4  |       |  3/5  |
       |________     _____|_____    ___|___      ___|___   __|__   __|__     |       |
       |        |   |           |  |       |    |       | |     | |     |
       E 1/(-2) |   |   2/(-1)  |  |  3/1  |    |  3/2  | | 4/5 | | 5/7 |
       |___     |   |           |  |       |    |       | |     | |     |
       |   |
  The horizontal transection at the point marked C contains denominators -1, 0, 1, 3, 2, ..., which is the negation of row 0. Similarly, horizontal transections at D and E give negations of rows 1 and 2. The table may be extended upward, that is, to negative indices n. The horizontal transections at points B and A give negations of rows n = -1 and n = -2, with terms -1, 1, 2, 5, 3, 10, ... and -1, 2, 3, 7, 4, 13, .... This extension is compatible with the recurrence on rows and with the arithmetic progression property of columns. (End)

Prog

(SageMath)
def T(rw, cl):              # Works for all integer rw.
    curCl = Integer(cl)
    curRw = Integer(rw)
    indList = []
    while curCl > 0:
        nextRw = curCl.nbits()
        indList.append(curRw - nextRw)
        curCl = curCl - 2^(nextRw-1)
        curRw = nextRw
    co1 = 1
    co0 = 1
    for i in range(len(indList)-1, -1, -1):
        newCo1 = co1 * (indList[i]+2) - co0
        co0 = newCo1 + co1
        co1 = newCo1
    return(co1)            # William P. Orrick, May 14 2026

Crossrefs

Cf. A049456, A070878, A070879, A049455, A002487.

Negations of rows: A156140, A321154.

Sequence in context: A394756 A359867 A021335 * A238133 A317922 A337320

Adjacent sequences: A089592 A089593 A089594 * A089596 A089597 A089598

Keyword

sign,tabl

Author

Philippe Deléham, Dec 30 2003

Status

approved