A372398 Numbers occurring exactly twice in Hofstadter G/H-like sequence H_5 (A005376).

1, 6, 7, 9, 12, 16, 21, 26, 27, 32, 33, 35, 40, 41, 43, 46, 51, 52, 54, 57, 61, 66, 67, 69, 72, 76, 81, 86, 87, 89, 92, 96, 101, 106, 107, 112, 113, 115, 118, 122, 127, 132, 133, 138, 139, 141, 146, 147, 149, 152, 156, 161, 166, 167, 172, 173, 175, 180, 181

Offset: 1

Author

A.H.M. Smeets, Apr 29 2024

Keywords

• nonn

Comments

Also first prepending column of the 5-Zeckendorf array (see Ericksen and Anderson).

N. J. A. Sloane observed already the relation between Hofstadter G/H-like sequences H_k and k-Zeckendorf arrays in May 2003, at least for k = 3 (see formula section and history of A005374). First observation most probably by Diego Torres, Nov 2002, relating the Hofstadter G/H-like sequences H_k with the k-Zeckendorf arrays and Lamé sequences of order k (see comments in A005375 and A005376).

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..20000
• Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012.

Crossrefs

Cf. A005374, A005376.

Numbers occurring exactly twice in Hofstadter G/H like sequence H_k: A000291 (k=2), A005374 (k=3), A372397 (k=4), this sequence (k=5).

Programs

Python

def H(n,k):
    if n == 0:
        return 0
    else:
        i, x = 0, n-1
        while i < k:
            i, x = i+1, H(x,k)
        return n-x
n, nn = 0, 0
while n < 59:
    if nn == 0:
        Hno = H(nn,5)
    else:
        Hnn = H(nn,5)
        if Hnn == Hno:
            n += 1
            print(Hnn, end = ", ")
        Hno = Hnn
    nn += 1