A005376 -id:A005376 - cp's OEIS frontend

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

A.H.M. Smeets, Apr 29 2024

nonn

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).

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.

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).

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

A005375 a(0) = 0; a(n) = n - a(a(a(a(n-1)))) for n > 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53

Offset: 0

N. J. A. Sloane

Rule for n-th term: a(n) = An, where An denotes the Lamé antecedent to (or right shift of) n, which is found by replacing each Lm(i) (Lm(n) = Lm(n-1) + Lm(n-4): A003269) in the Zeckendorffian expansion (obtained by repeatedly subtracting the largest Lamé number you can until nothing remains) by Lm(i-1) (A1=1). For example: 58 = 50 + 7 + 1, so a(58)= 36 + 5 + 1 = 42. - Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002
a(A194081(n)) = n and a(m) <> n for m < A194081(n). - Reinhard Zumkeller, Aug 17 2011
From Pierre Letouzey, Feb 20 2025: (Start)
For all n >= 0, A005374(n) <= a(n) <= A005376(n) as proved in Letouzey-Li-Steiner link.
Last equality A005374(n) = a(n) for n = 18; last equality a(n) = A005376(n) for n = 25.
For all n >= 0, |a(n) - c*n| < 1.998 where c is the positive real root of x^4 + x - 1 = 0, c = 0.724491959000515611588372282... Proved in Letouzey link. (End)

D. Hofstadter, "Goedel, Escher, Bach", p. 137.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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.
Nick Hobson, Python program for this sequence
Pierre Letouzey, S. Li and W. Steiner, Pointwise order of generalized Hofstadter functions G,H and beyond, arXiv:2410.00529 [cs.DM], 2024.
Pierre Letouzey, Generalized Hofstadter functions G,H and beyond: numeration systems and discrepancy arXiv:2502.12615 [cs.DM], 2025.
Index entries for Hofstadter-type sequences
Index entries for sequences from "Goedel, Escher, Bach"

a005375 n = a005375_list !! n
a005375_list =  0 : 1 : zipWith (-)
   [2..] (map a005375 (map a005375 (map a005375 (tail a005375_list))))
-- Reinhard Zumkeller, Aug 17 2011

H:=proc(n) option remember; if n=1 then 1 else n-H(H(H(H(n-1)))); fi; end proc;

a[0]:= 0; a[n_]:= a[n]= a[n] = n - a[a[a[a[n-1]]]]; Table[a[n], {n, 0, 73}] (* Alonso del Arte, Aug 17 2011 *)

@CachedFunction # a = A005375
def a(n): return 0 if (n==0) else n - a(a(a(a(n-1))))
[a(n) for n in range(101)] # G. C. Greubel, Nov 14 2022

a(n) = floor(c*n) + (-1) or 0 or 1 or 2, where c is the positive real root of x^4+x-1 = 0, c=0.724491959000515611588372282... (Conjectured with just 0 or 1 by Benoit Cloitre, Nov 05 2002; fixed and proved by Letouzey, see Letouzey link]. NB: see for instance a(120) = 88 for a difference of 2 and a(243) = 175 for a difference of -1). - Pierre Letouzey, Feb 20 2025

a(n + a(a(a(n)))) = n (proved in Letouzey-Li-Steiner link). - Pierre Letouzey, Feb 20 2025

More terms from James Sellers, Jul 12 2000

A100721 a(n) = n - a(a(a(a(a(a(n-1)))))), a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 34, 35, 35, 36, 37, 38, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 50, 51, 51, 52, 53, 53, 54, 55, 56, 56, 57, 58, 59

Offset: 0

N. J. A. Sloane, Dec 12 2004

nonn

From Pierre Letouzey, Mar 06 2025: (Start)
For all n >= 0, A005376(n) <= a(n) as proved in Letouzey-Li-Steiner link. Last equality a(n) = A005376(n) for n = 33. Moreover a(n) <= b(n) for all sequences b also defined by b(0)=0 and then b(n)=n-b(...b(n-1)...) with more than 6 nested recursive calls.
a(n) = c*n + O(n^d), where c is the real root of x^6+x-1 = 0, c=0.7780895986786012... and d=0.1287... Proved in Letouzey link. See also Dilcher 1993. (End)

Karl Dilcher, On a class of iterative recurrence relations, in G. E. Bergum, A. N. Philippou, and A. F. Horadam, editors, Applications of Fibonacci Numbers, vol. 5, p. 143-158, Springer, 1993.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Pierre Letouzey, Generalized Hofstadter functions G,H and beyond: numeration systems and discrepancy, arXiv:2502.12615 [cs.DM], 2025.
Pierre Letouzey, Shuo Li, and Wolfgang Steiner, Pointwise order of generalized Hofstadter functions G, H and beyond, arXiv:2410.00529 [cs.DM], 2024. See p. 1.

Cf. A005206, A005374, A005375, A005376.

H:=proc(n) option remember; if n=0 then 0 else n-H(H(H(H(H(H(n-1)))))); fi; end proc;

a[0]= 0; a[n_]:= a[n]= n - a[a[a[a[a[a[n-1]]]]]]; Table[ a[n], {n, 75}] (* Robert G. Wilson v, Dec 16 2004 *)

@CachedFunction # a = A100721
def a(n): return 0 if (n==0) else n - a(a(a(a(a(a(n-1))))))
[a(n) for n in range(1,100)] # G. C. Greubel, Nov 16 2022

a(n + a(a(a(a(a(n)))))) = n (proved in Letouzey-Li-Steiner link). - Pierre Letouzey, Mar 06 2025

a(0)=0 inserted by Pierre Letouzey, Mar 07 2025

A372397 Numbers occurring exactly twice in Hofstadter G/H-like sequence H_4 (A005375).

Original entry on oeis.org

1, 5, 6, 8, 11, 15, 19, 20, 24, 25, 27, 31, 32, 34, 37, 41, 42, 44, 47, 51, 55, 56, 58, 61, 65, 69, 70, 74, 75, 77, 80, 84, 88, 89, 93, 94, 96, 100, 101, 103, 106, 110, 114, 115, 119, 120, 122, 126, 127, 129, 132, 136, 137, 139, 142, 146, 150, 151, 155, 156

Offset: 1

A.H.M. Smeets, Apr 29 2024

nonn

Also first prepending column of the 4-Zeckendorf array (see Ericksen and Anderson).

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.

Cf. A005374, A005375.

Numbers occurring exactly twice in Hofstadter G/H like sequence H_k: A000291 (k=2), A005374 (k=3), this sequence (k=4), A372398 (k=5).

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 < 50:
    if nn == 0:
        Hno = H(nn,4)
    else:
        Hnn = H(nn,4)
        if Hnn == Hno:
            n += 1
            print(Hnn, end = ", ")
        Hno = Hnn
    nn += 1

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

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A005375 a(0) = 0; a(n) = n - a(a(a(a(n-1)))) for n > 0.

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A100721 a(n) = n - a(a(a(a(a(a(n-1)))))), a(0)=0.

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A372397 Numbers occurring exactly twice in Hofstadter G/H-like sequence H_4 (A005375).

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