A385455 -id:A385455 - cp's OEIS frontend

A385436 Tribonacci array of the second kind, read by upward antidiagonals.

0, 2, 1, 4, 5, 3, 6, 8, 10, 7, 9, 12, 16, 20, 14, 11, 18, 23, 31, 38, 27, 13, 21, 34, 44, 58, 71, 51, 15, 25, 40, 64, 82, 108, 132, 95, 17, 29, 47, 75, 119, 152, 200, 244, 176, 19, 32, 54, 88, 139, 220, 281, 369, 450, 325, 22, 36, 60, 101, 163, 257, 406, 518, 680

Offset: 1

A.H.M. Smeets, Jun 28 2025

The array is, as a sequence, a permutation of the nonnegative integers; however it does not satisfy the conditions for interspersion and dispersion as given by Eric Weisstein's World of Mathematics. However, when all terms are increased by 1, it does satisfy the conditions for interspersion and dispersion!
Rows satisfy the recurrence: T(m,k) = 2*T(m,k-1) - T(m,k-4) for all k>4.
This array belongs to a family of Wythoff like arrays, based on binary number representations like the greedy and lazy Fibonacci number representations (see A035513 and A372501 for arrays), greedy and lazy Narayana number representations (A136189 for the array related to greedy representation).
The array is related to the lazy tribonacci number representation A352103. The first column lists the even numbers, i.e., for wich 0 suffix A352103(T(m,1)). The odd numbers are represented in the columns k > 1: A352103(T(m,k)) = A352103(T(m,1)) + 1^(k-1). Here + stands for concatenation and ^ stands for repeated concatenation.

			Array including some prepended columns (p = 1..4):
  p=4 p=3 p=2 p=1 | k=1 k=2 k=3  k=4  k=5  k=6  k=7   k=8   k=9  k=10
   -2  -1  -1  -1 |   0   1   3    7   14   27   51    95   176   325
   -2  -1   0   0 |   2   5  10   20   38   71  132   244   450   829
   -2   0   0   1 |   4   8  16   31   58  108  200   369   680
   -2   0   1   2 |   6  12  23   44   82  152  281   518
   -1   0   2   4 |   9  18  34   64  119  220  406   748
   -1   1   2   5 |  11  21  40   75  139  257  474   873
   -1   1   3   6 |  13  25  47   88  163  301  555  1022
   -1   1   4   7 |  15  29  54  101  187  345  636  1171
   -1   2   4   8 |  17  32  60  112  207  382  704  1296
   -1   2   5   9 |  19  36  67  125
    0   2   6  11 |  22  42  78  145
Each row of the array satisfies the recurrence relation T(m,k) = 2*T(m,k-1) - T(m,k-4); from this, the prepended columns are obtained by rowwise backward recursion.

A.H.M. Smeets, Table of n, a(n) for n = 1..20100 (first 200 antidiagonals).
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Eric Weisstein's World of Mathematics, Interspersion.
Eric Weisstein's World of Mathematics, Sequence Dispersion.

Cf. A035513, A136189, A372501, A027084 (m=1), A351631 (k=1), A352103.

Prepended columns: A385455 (p=1), A385532 (p=2), A385533 (p=3).

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
def From_111_Zeck(s):
    f0, f1, f2, i, n = 1, 1, 0, len(s), 0
    while i > 0:
        i -= 1
        f0, f1, f2, n = f0+f1+f2, f0, f1, n+int(s[i])*f0
    return n
d, a, n, c1 = 0, 0, 0, []
while d < 11:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n, d = n+1, d+1
        print(a, end = ", ")
        i, c1, p1 = d-1, c1+[s], ""
        while i > 0:
            n, i, p1 = n+1, i-1, p1+"1"
            print(From_111_Zeck(c1[i]+p1), end = ", ")
    a += 1

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A385436 Tribonacci array of the second kind, read by upward antidiagonals.

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