A063882 -id:A063882 - cp's OEIS frontend

A317686 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A063882.

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

Offset: 1

Altug Alkan, Aug 04 2018

nonn

This sequence hits every positive integer and it has a fractal-like structure, see scatterplot of 2*n-3*a(n) in Links section.

Let b(1) = b(2) = b(3) = b(4) = 1; for n >= 5, b(n) = b(t(n)) + b(n-t(n)) where t = A063882. Observe the symmetric relation between this sequence (a(n)) and b(n) thanks to plots of a(n)-2*n/3 and b(n)-n/3 in Links section. Note that a(n) + b(n) = n for n >= 2.

Altug Alkan, Scatterplot of 2*n-3*a(n) for n <= 36000
Altug Alkan, Scatterplots of a(n)-2*n/3 and b(n)-n/3 for n <= 36000

Cf. A063882, A317648.

b:= proc(n) option remember; `if`(n<5, 1,
      b(n-b(n-1)) +b(n-b(n-4)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      a(b(n)) +a(n-b(n)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 05 2018

b[n_] := b[n] = If[n < 5, 1, b[n - b[n - 1]] + b[n - b[n - 4]]];
a[n_] := a[n] = If[n < 3, 1, a[b[n]] + a[n - b[n]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

t=vector(99); t[1]=t[2]=t[3]=t[4]=1; for(n=5, #t, t[n] = t[n-t[n-1]]+t[n-t[n-4]]); a=vector(99); a[1]=a[2]=1; for(n=3, #a, a[n] = a[t[n]]+a[n-t[n]]); a

a(n+1) - a(n) = 0 or 1 for all n >= 1.

A132157 a(n) = number of times n occurs in A063882.

Original entry on oeis.org

4, 1, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2

Offset: 1

N. J. A. Sloane, Nov 06 2007

nonn

a(A202016(n)) = 1. [Reinhard Zumkeller, Dec 08 2011]

N. J. A. Sloane, Table of n, a(n) for n = 1..200 [Taken from the Balamohan et al. reference]
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Index entries for Hofstadter-type sequences

import Data.List (group)
a132157 n = a132157_list !! (n-1)
a132157_list = (map length) (group a063882_list)
-- Reinhard Zumkeller, Dec 08 2011

A202016 Numbers occurring only once in A063882.

Original entry on oeis.org

2, 3, 4, 7, 10, 13, 16, 19, 24, 28, 31, 36, 40, 46, 50, 54, 60, 64, 70, 74, 78, 84, 90, 94, 98, 104, 107, 110, 114, 118, 122, 126, 132, 138, 142, 146, 152, 155, 158, 162, 166, 172, 178, 182, 186, 192, 195, 198, 202, 206, 212, 216, 219, 224, 227, 234, 238

Offset: 1

Reinhard Zumkeller, Dec 08 2011

nonn

A132157(a(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndices)
a202016 n = a202016_list !! (n-1)
a202016_list = map (+ 1) $ elemIndices 1 a132157_list

A132174 Index of starting position of n-th generation of terms in A063882.

Original entry on oeis.org

1, 5, 10, 21, 44, 92, 189, 385, 778, 1565, 3141, 6294, 12602, 25219, 50454, 100926, 201871, 403763, 807548, 1615119, 3230263, 6460552, 12921132, 25842293, 51684616, 103369264, 206738561, 413477157, 826954350, 1653908737, 3307817513, 6615635066, 13231270174

Offset: 1

N. J. A. Sloane, Nov 07 2007

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1002
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Index entries for Hofstadter-type sequences
Index entries for linear recurrences with constant coefficients, signature (3, -2, 0, 0, 1, -3, 2).

from _future_ import division
def A132174(n):
    if n == 1:
        return 1
    if n == 2:
        return 5
    h, m = divmod(n - 3, 5)
    return (382*2**(5*h + m)-10*2**m)//31- 7*h - m -(1 if m==3 else (-1 if m==4 else 2)) # Chai Wah Wu, May 17 2017

From Chai Wah Wu, May 17 2017: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-5) - 3*a(n-6) + 2*a(n-7) for n > 8.
G.f.: x*(-5*x^7 + x^6 - x^5 - x^4 - x^3 + 3*x^2 - 2*x - 1)/((x - 1)^2*(2*x - 1)*(x^4 + x^3 + x^2 + x + 1)). (End)

More terms from Chai Wah Wu, May 17 2017

A202014 Smallest m such that A063882(m) = n.

Original entry on oeis.org

1, 5, 6, 7, 8, 10, 12, 13, 15, 17, 18, 21, 23, 24, 26, 28, 29, 32, 34, 35, 37, 39, 42, 44, 45, 47, 49, 51, 52, 55, 57, 58, 60, 62, 65, 67, 68, 70, 72, 74, 75, 78, 80, 82, 85, 87, 88, 90, 92, 94, 95, 98, 100, 102, 103, 105, 107, 109, 112, 114, 115, 117, 119

Offset: 1

Reinhard Zumkeller, Dec 08 2011

nonn

A063882(a(n)) = n and A063882(m) < n for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a202014 n = (fromJust $ elemIndex n a063882_list) + 1

A319020 Let b_i(k) = 1 for k <= i; for n > i, b_i(n) = b_i(t(n)) + b_i(n-t(n)) where t = A063882. a(n) = 3b_2(n)-2n if n is even, a(n) = 3*b_4(n)-n if n is odd.

Original entry on oeis.org

2, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, 0, 2, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, 0, -1, -1, 0, -2, 1, 0, -1, 2, -3, 1, 1, -3, 2, -1, 3, -2, 1, 0, -1, -1, 0, 1, -2, 0, 2, -1, 0, -2, 1, 0, -1, -1, 0, -2, 1, 0, -1, 2, 0, 1, -2, 3, -1, -1, 3, -2, 1, -3, 2, -1, 0, 1, -2, 0, 2, -4, 3, -2, 4, -3, 2, -1, 0, -2

Offset: 1

Altug Alkan, Sep 08 2018

Altug Alkan, Table of n, a(n) for n = 1..9216

Cf. A063882, A317686, A317754, A317854.

t=f=g=vector(200); t[1]=t[2]=t[3]=t[4]=1; for(n=5, #t, t[n] = t[n-t[n-1]]+t[n-t[n-4]]); f[1]=f[2]=1; for(n=3, #f, f[n] = f[t[n]]+f[n-t[n]]); g[1]=g[2]=g[3]=g[4]=1; for(n=5, #g, g[n] = g[t[n]]+g[n-t[n]]); vector(200, n, if(n%2==0, 3*f[n]-2*n,3*g[n]-n))

A129632 Partial sums of A063882.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 18, 23, 29, 35, 42, 50, 58, 67, 76, 86, 97, 108, 119, 131, 143, 156, 170, 184, 199, 214, 230, 247, 264, 281, 299, 317, 336, 356, 376, 397, 418, 440, 462, 484, 507, 530, 554, 579, 604, 630, 656, 683, 710, 738, 767, 796, 825, 855, 885, 916, 948, 980, 1013

Offset: 1

Gary W. Adamson, Nov 08 2007

nonn

A132175 Index of end of n-th generation of terms in A063882.

Original entry on oeis.org

4, 9, 20, 43, 91, 188, 384, 777, 1564, 3140, 6293, 12601, 25218, 50453, 100925, 201870, 403762, 807547

Offset: 1

N. J. A. Sloane, Nov 07 2007

nonn

B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Index entries for Hofstadter-type sequences

A132176 Value of A063882 at start of n-th generation of terms.

Original entry on oeis.org

1, 2, 6, 12, 24, 49, 98, 197, 394, 788, 1577, 3154, 6309, 12618, 25236, 50473, 100946, 201893

Offset: 1

N. J. A. Sloane, Nov 07 2007

nonn

B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Index entries for Hofstadter-type sequences

A132177 Value of A063882 at end of n-th generation of terms.

Original entry on oeis.org

1, 5, 11, 23, 48, 97, 196, 393, 787, 1576, 3153, 6308, 12617, 25235, 50472, 100945, 201892, 403785

Offset: 1

N. J. A. Sloane, Nov 07 2007

nonn

B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Index entries for Hofstadter-type sequences

cp's OEIS Frontend

A317686 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A063882.

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A132157 a(n) = number of times n occurs in A063882.

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A202016 Numbers occurring only once in A063882.

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A132174 Index of starting position of n-th generation of terms in A063882.

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A202014 Smallest m such that A063882(m) = n.

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Haskell

A319020 Let b_i(k) = 1 for k <= i; for n > i, b_i(n) = b_i(t(n)) + b_i(n-t(n)) where t = A063882. a(n) = 3*b_2(n)-2*n if n is even, a(n) = 3*b_4(n)-n if n is odd.

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PARI

A129632 Partial sums of A063882.

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A132175 Index of end of n-th generation of terms in A063882.

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A132176 Value of A063882 at start of n-th generation of terms.

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A132177 Value of A063882 at end of n-th generation of terms.

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A319020 Let b_i(k) = 1 for k <= i; for n > i, b_i(n) = b_i(t(n)) + b_i(n-t(n)) where t = A063882. a(n) = 3b_2(n)-2n if n is even, a(n) = 3*b_4(n)-n if n is odd.