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

%I A385436 #30 Jul 09 2025 10:17:25
%S A385436 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,
%T A385436 71,51,15,25,40,64,82,108,132,95,17,29,47,75,119,152,200,244,176,19,
%U A385436 32,54,88,139,220,281,369,450,325,22,36,60,101,163,257,406,518,680
%N A385436 Tribonacci array of the second kind, read by upward antidiagonals.
%C A385436 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!
%C A385436 Rows satisfy the recurrence: T(m,k) = 2*T(m,k-1) - T(m,k-4) for all k>4.
%C A385436 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).
%C A385436 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.
%H A385436 A.H.M. Smeets, <a href="/A385436/b385436.txt">Table of n, a(n) for n = 1..20100</a> (first 200 antidiagonals).
%H A385436 Larry Ericksen and Peter G. Anderson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/50-1/EricksenAnderson.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
%H A385436 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly 33 (1995) 3-8.
%H A385436 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Interspersion.html">Interspersion</a>.
%H A385436 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SequenceDispersion.html">Sequence Dispersion</a>.
%e A385436 Array including some prepended columns (p = 1..4):
%e A385436   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
%e A385436    -2  -1  -1  -1 |   0   1   3    7   14   27   51    95   176   325
%e A385436    -2  -1   0   0 |   2   5  10   20   38   71  132   244   450   829
%e A385436    -2   0   0   1 |   4   8  16   31   58  108  200   369   680
%e A385436    -2   0   1   2 |   6  12  23   44   82  152  281   518
%e A385436    -1   0   2   4 |   9  18  34   64  119  220  406   748
%e A385436    -1   1   2   5 |  11  21  40   75  139  257  474   873
%e A385436    -1   1   3   6 |  13  25  47   88  163  301  555  1022
%e A385436    -1   1   4   7 |  15  29  54  101  187  345  636  1171
%e A385436    -1   2   4   8 |  17  32  60  112  207  382  704  1296
%e A385436    -1   2   5   9 |  19  36  67  125
%e A385436     0   2   6  11 |  22  42  78  145
%e A385436 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.
%o A385436 (Python)
%o A385436 def ToDual_111_Zeck(n):
%o A385436     if n == 0:
%o A385436         return "0"
%o A385436     f0, f1, f2, sf = 1, 0, 0, 0
%o A385436     while n > sf:
%o A385436         f0, f1, f2 = f0+f1+f2, f0, f1
%o A385436         sf += f0
%o A385436     r, s = sf-n, "1"
%o A385436     while f0 > 1:
%o A385436         f0, f1, f2 = f1, f2, f0-f1-f2
%o A385436         r, s = r%f0, s+str(1-r//f0)
%o A385436     return s
%o A385436 def From_111_Zeck(s):
%o A385436     f0, f1, f2, i, n = 1, 1, 0, len(s), 0
%o A385436     while i > 0:
%o A385436         i -= 1
%o A385436         f0, f1, f2, n = f0+f1+f2, f0, f1, n+int(s[i])*f0
%o A385436     return n
%o A385436 d, a, n, c1 = 0, 0, 0, []
%o A385436 while d < 11:
%o A385436     s = ToDual_111_Zeck(a)
%o A385436     if s[len(s)-1] == "0": # == even
%o A385436         n, d = n+1, d+1
%o A385436         print(a, end = ", ")
%o A385436         i, c1, p1 = d-1, c1+[s], ""
%o A385436         while i > 0:
%o A385436             n, i, p1 = n+1, i-1, p1+"1"
%o A385436             print(From_111_Zeck(c1[i]+p1), end = ", ")
%o A385436     a += 1
%Y A385436 Cf. A035513, A136189, A372501, A027084 (m=1), A351631 (k=1), A352103.
%Y A385436 Prepended columns: A385455 (p=1), A385532 (p=2), A385533 (p=3).
%K A385436 nonn,tabl
%O A385436 1,2
%A A385436 _A.H.M. Smeets_, Jun 28 2025