A317686 -id:A317686 - cp's OEIS frontend

A317754 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A004001. a(n) = 2*b(n) - n.

1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, -1, 0, -1, -2, -3, -4, -3, -2, -1, 0, -1, -2, -3, -4, -5, -6, -5, -4, -3, -2, -1, 0, -1, -2, -3, -4, -3, -2, -1, 0, -1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 1

Altug Alkan, Aug 06 2018

Altug Alkan, Table of n, a(n) for n = 1..32768

Cf. A004001, A004074, A317648, A317742.

t:= proc(n) option remember; `if`(n<3, 1,
      t(t(n-1)) +t(n-t(n-1)))
    end:
b:= proc(n) option remember; `if`(n<3, 1,
      n -b(t(n)) -b(n-t(n)))
    end:
seq(2*b(n)-n, n=1..100); # after Alois P. Heinz at A317686

Block[{t = NestWhile[Function[{a, n}, Append[a, a[[a[[-1]] ]] + a[[-a[[-1]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Last@ # < 60 &], b}, b = NestWhile[Function[{b, n}, Append[b, n - b[[t[[n]] ]] - b[[-t[[n]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Length@ # < Length@ t &]; Array[2 b[[#]] - # &, Length@ b] ] (* Michael De Vlieger, Aug 07 2018 *)

t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = t[n-t[n-1]]+t[t[n-1]]); b=vector(99); b[1]=b[2]=1; for(n=3, #b, b[n] = n-b[t[n]]-b[n-t[n]]); vector(99, k, 2*b[k]-k)

A317648 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A004001.

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Aug 02 2018

This sequence hits every positive integer.

Let b(1) = b(2) = b(3) = 1; for n >= 4, b(n) = b(t(n)) + b(n-t(n)) where t = A004001. Observe the symmetric relation between this sequence (a(n)) and b(n) thanks to line plots of a(n)-n/2 and b(n)-n/2 in Links section.

Altug Alkan, Table of n, a(n) for n = 1..65536
Altug Alkan, Line plot of a(n)-n/2 for n <= 2^17
Altug Alkan, Line plots of A004001(n)-n/2 and a(n)-n/2 for n <= 2^14
Altug Alkan, Line plots of a(n)-n/2 and b(n)-n/2 for n <= 2^11

Cf. A004001, A317686.

b:= proc(n) option remember; `if`(n<3, 1,
      b(b(n-1)) +b(n-b(n-1)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      a(b(n)) +a(n-b(n)))
    end:
seq(a(n), n=1..100); # after Alois P. Heinz at A317686

t[1] = 1; t[2] = 1; t[n_] := t[n] = t[t[n-1]] + t[n - t[n-1]];
a[1] = a[2] = 1; a[n_] := a[n] = a[t[n]] + a[n - t[n]];
Array[a, 100] (* Jean-François Alcover, Nov 01 2020 *)

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

A317854 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A287422. a(n) = 2*b(n) - n.

Original entry on oeis.org

1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 0, 1, 0, -1, 0, -1, -2, -3, -4, -3, -2, -1, -2, -3, -2, -3, -4, -5, -6, -5, -4, -3, -2, -3, -2, -1, -2, -3, -4, -3, -2, -1, 0, -1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 4, 5

Offset: 1

Altug Alkan, Aug 09 2018

A different version of A317742. Similar to A317754.

Altug Alkan, Table of n, a(n) for n = 1..32768
Altug Alkan, Line plot of a(n) for n <= 2^17
Rémy Sigrist, C++ program for A317854

Cf. A287422, A317742, A317754.

t:= proc(n) option remember; `if`(n<3, 1,
      n -t(t(n-1)) -t(n-t(n-1)))
    end:
b:= proc(n) option remember; `if`(n<3, 1,
      n -b(t(n)) -b(n-t(n)))
    end:
seq(2*b(n)-n, n=1..100); # after Alois P. Heinz at A317686

t[1]=t[2]=1; t[n_] := t[n] = n - t[t[n-1]] - t[n - t[n-1]]; b[1]=b[2]=1; b[n_] := b[n] = n - b[t[n]] - b[n - t[n]]; a[n_] := 2*b[n] - n; Array[a, 95] (* Giovanni Resta, Aug 14 2018 *)

t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = n-t[n-t[n-1]]-t[t[n-1]]); b=vector(99); b[1]=b[2]=1; for(n=3, #b, b[n] = n-b[t[n]]-b[n-t[n]]); vector(99, k, 2*b[k]-k)
(C++) See Links section.

abs(a(n)) = A317742(n).

A317742 Let b(1) = b(2) = 1; for n >= 3, b(n) = b(t(n)) + b(n-t(n)) where t = A287422. a(n) = 2*b(n) - n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 5, 6, 5, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 8, 9, 10

Offset: 1

Altug Alkan, Aug 05 2018

This sequence has fractal-like structure as A004074, although there are substantial differences of this sequence based on slow A287422 and b(n) sequences. See plots of this sequence and A004074 in Links section.

Altug Alkan, Table of n, a(n) for n = 1..16384
Altug Alkan, Line plots of a(n) and A004074(n) for n <= 2^15

Cf. A004074, A287422, A317648, A317686.

t:= proc(n) option remember; `if`(n<3, 1,
      n -t(t(n-1)) -t(n-t(n-1)))
    end:
b:= proc(n) option remember; `if`(n<3, 1,
      b(t(n)) +b(n-t(n)))
    end:
seq(2*b(n)-n, n=1..100); # after Alois P. Heinz at A317686

Block[{t = NestWhile[Function[{a, n}, Append[a, n - a[[a[[-1]] ]] - a[[-a[[-1]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Last@ # < 10^2 &], b}, b = NestWhile[Function[{b, n}, Append[b, b[[t[[n]] ]] + b[[-t[[n]] ]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, Last@ # < Max@ t &]; Array[2 b[[#]] - # &, Length@ b] ] (* Michael De Vlieger, Aug 07 2018 *)
t[n_] := t[n] = If[n<3, 1, n - t[t[n-1]] - t[n - t[n-1]]]; b[n_] := b[n] = If[n<3, 1, b[t[n]] + b[n - t[n]]]; Table[2*b[n] - n, {n, 106}] (* Giovanni Resta, Aug 14 2018 *)

t=vector(199); t[1]=t[2]=1; for(n=3, #t, t[n] = n-t[n-t[n-1]]-t[t[n-1]]); b=vector(199); b[1]=b[2]=1; for(n=3, #b, b[n] = b[t[n]]+b[n-t[n]]); vector(199, k, 2*b[k]-k)

A318056 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n-1)) where t = A004001. a(n) = 2*b(n) - n.

Original entry on oeis.org

1, 0, -1, 0, 1, 0, 1, 0, -3, -2, -3, -2, -1, 0, -1, 0, 5, 4, 5, 4, 5, 2, 1, 2, 3, 4, 3, 0, 1, 0, 1, 0, -11, -10, -11, -10, -11, -10, -7, -6, -7, -6, -9, -10, -9, -8, -7, -8, -3, -4, -3, -2, -1, -2, -5, -4, -5, -4, -1, 0, -1, 0, -1, 0, 21, 20, 21, 20, 21, 20, 21, 16, 15, 16, 15, 16, 19, 20, 19, 20

Offset: 1

Altug Alkan, Aug 14 2018

Altug Alkan, Table of n, a(n) for n = 1..32768
Altug Alkan, Line plot of a(n) for n <= 2^17

Cf. A004001, A317754, A317854.

t:= proc(n) option remember; `if`(n<3, 1,
      t(t(n-1)) +t(n-t(n-1)))
    end:
b:= proc(n) option remember; `if`(n<3, 1,
      n -b(t(n)) -b(n-t(n-1)))
    end:
seq(2*b(n)-n, n=1..100); # after Alois P. Heinz at A317686

t[1]=t[2]=1; t[n_] := t[n] = t[t[n-1]] + t[n - t[n-1]]; b[1]=b[2]=1; b[n_] := b[n] = n - b[t[n]] - b[n - t[n-1]]; a[n_] := 2*b[n] - n; Array[a, 95] (* after Giovanni Resta at A317854 *)

t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = t[n-t[n-1]]+t[t[n-1]]); b=vector(99); b[1]=b[2]=1; for(n=3, #b, b[n] = n-b[t[n]]-b[n-t[n-1]]);vector(99, k, 2*b[k]-k)

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))

A300623 Let b(1) = 1; for n >= 2, b(n) = n - b(t(n)) - b(n-t(n-1)) where t = A302128. a(n) = 2*b(n) - n.

Original entry on oeis.org

1, -2, 1, 0, 1, -2, -1, -2, 1, 2, -1, 0, 1, 0, 1, -2, -1, 2, 3, 2, 1, -2, -1, -4, -3, 2, 3, 0, 1, -4, -3, -4, -3, -2, 1, 2, -1, 0, 5, 6, 3, 4, -5, -4, -7, -6, -3, -2, -3, 2, 3, 2, 3, 2, 1, 0, 1, -2, -1, 4, 5, 2, 3, -8, -7, -10, -9, -4, -3, -6, -5, 8, 9, 6, 7, 10, 11, 10, 5, 6, 5, 6, -1, 0, -1, 0, -1, 0, 1, -2, -1, -4, -3

Offset: 1

Altug Alkan, Aug 14 2018

Sequence has a fractal-like structure. Fibonacci numbers (A000045) are determinative for the generational boundaries.

Altug Alkan, Table of n, a(n) for n = 1..28657
Altug Alkan, Line plot of a(n) for n <= 28657
Altug Alkan, Scatterplot of a(n) for n <= 75025

Cf. A302128, A318056.

t:= proc(n) option remember; `if`(n<4, 1,
      t(t(n-2)) +t(n-t(n-1)))
    end:
b:= proc(n) option remember; `if`(n<2, 1,
      n -b(t(n)) -b(n-t(n-1)))
    end:
seq(2*b(n)-n, n=1..100); # after Alois P. Heinz at A317686

t[1]=t[2]=t[3]=1; t[n_] := t[n] = t[t[n-2]] + t[n - t[n-1]]; b[1]=1; b[n_] := b[n] = n - b[t[n]] - b[n - t[n-1]]; a[n_] := 2*b[n] - n; Array[a, 95] (* after Giovanni Resta at A317854 *)

t=vector(99); t[1]=t[2]=t[3]=1; for(n=4, #t, t[n] = t[n-t[n-1]]+t[t[n-2]]); b=vector(99); b[1]=1; for(n=2, #b, b[n] = n-b[t[n]]-b[n-t[n-1]]); vector(99,k, 2*b[k]-k)

A309430 a(n) = a(n-a(n-1)) + a(n-a(n-4)), with a(n) = ceiling(2*n/3) for n <= 7.

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Aug 01 2019

A slow solution to Hofstadter V recurrence.

Numbers k such that a(k) < A063882(k) are 1654, 1721, 1925, ...

Cf. A063882, A317686, A319020.

q=vector(100); for(n=1, 7, q[n]=ceil(2*n/3));for(n=8, #q, q[n]=q[n-q[n-1]] +q[n-q[n-4]]); q

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

cp's OEIS Frontend

A317754 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A004001. a(n) = 2*b(n) - n.

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A317648 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A004001.

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A317854 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n)) where t = A287422. a(n) = 2*b(n) - n.

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A317742 Let b(1) = b(2) = 1; for n >= 3, b(n) = b(t(n)) + b(n-t(n)) where t = A287422. a(n) = 2*b(n) - n.

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A318056 Let b(1) = b(2) = 1; for n >= 3, b(n) = n - b(t(n)) - b(n-t(n-1)) where t = A004001. a(n) = 2*b(n) - n.

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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) = 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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A300623 Let b(1) = 1; for n >= 2, b(n) = n - b(t(n)) - b(n-t(n-1)) where t = A302128. a(n) = 2*b(n) - n.

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A309430 a(n) = a(n-a(n-1)) + a(n-a(n-4)), with a(n) = ceiling(2*n/3) for n <= 7.

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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.