A244477 -id:A244477 - cp's OEIS frontend

A304490 a(1) = a(2) = a(3) = 1, a(4) = 5, a(5) = 6, a(6) = 4; a(n) = a(n-a(n-2)) + a(n-a(n-4)) for n > 6.

1, 1, 1, 5, 6, 4, 5, 6, 6, 9, 10, 5, 6, 12, 12, 15, 16, 5, 6, 18, 18, 21, 22, 5, 6, 24, 24, 27, 28, 5, 6, 30, 30, 33, 34, 5, 6, 36, 36, 39, 40, 5, 6, 42, 42, 45, 46, 5, 6, 48, 48, 51, 52, 5, 6, 54, 54, 57, 58, 5, 6, 60, 60, 63, 64, 5, 6, 66, 66, 69, 70, 5, 6, 72, 72, 75, 76, 5, 6, 78, 78, 81, 82, 5, 6

Offset: 1

Altug Alkan, May 13 2018

A quasi-periodic solution to the recurrence a(n) = a(n-a(n-2)) + a(n-a(n-4)). Although A087777 and A240809 are highly chaotic, this sequence is completely predictable thanks to its initial conditions.

Index entries for linear recurrences with constant coefficients, signature (2, -3, 4, -5, 6, -5, 4, -3, 2, -1).

Cf. A087777, A240809, A244477, A296518, A296786.

q=vector(85); q[1]=1;q[2]=1;q[3]=1;q[4]=5;q[5]=6;q[6]=4; for(n=7, #q, q[n] = q[n-q[n-2]]+q[n-q[n-4]]); q

a(6*k) = 5, a(6*k+1) = 6, a(6*k+2) = a(6*k+3) = 6*k, a(6*k+4) = 6*k+3, a(6*k+5) = 6*k+4 for k > 1.
Conjectures from Colin Barker, May 14 2018: (Start)
G.f.: x*(1 - x + 2*x^2 + 2*x^3 + 2*x^5 - x^6 + 4*x^7 - 3*x^8 + x^9 - x^10 - 2*x^11 + 2*x^12 - x^13 + x^14 + x^15 - x^16) / ((1 - x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 5*a(n-4) + 6*a(n-5) - 5*a(n-6) + 4*a(n-7) - 3*a(n-8) + 2*a(n-9) - a(n-10) for n>17.
(End)

A304493 a(0) = 1; a(n) = a(n-a(floor(n/2))) + a(n-a(floor(n/4))).

Original entry on oeis.org

1, 2, 3, 5, 5, 8, 7, 11, 13, 12, 14, 18, 19, 20, 17, 19, 23, 24, 27, 28, 26, 31, 22, 27, 32, 34, 35, 37, 42, 46, 46, 45, 40, 40, 45, 40, 44, 48, 49, 55, 52, 56, 60, 65, 57, 64, 70, 72, 69, 70, 68, 64, 64, 67, 69, 67, 72, 71, 75, 80, 73, 79, 89, 84, 88, 94, 100, 94, 84, 96, 116, 117, 106, 116, 107, 106

Offset: 0

Ilya Gutkovskiy, May 13 2018

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for Hofstadter-type sequences

Cf. A002265, A004526, A005185, A083662, A244477.

f:= proc(n) option remember;
  procname(n-procname(floor(n/2)))+procname(n-procname(floor(n/4)))
end proc:
f(0):= 1;
map(f, [$0..100]); # Robert Israel, Dec 02 2019

a[n_] := a[n] = a[n - a[Floor[n/2]]] + a[n - a[Floor[n/4]]]; a[0] = 1; Table[a[n], {n, 0, 75}]

A304621 a(n) = 10 - n for 1 <= n <= 9. Thereafter a(n) = a(n-a(n-3)) + a(n-a(n-6)).

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 5, 7, 15, 8, 10, 12, 8, 7, 9, 14, 13, 24, 11, 13, 21, 17, 16, 9, 14, 16, 33, 17, 16, 30, 20, 28, 9, 20, 16, 42, 26, 25, 39, 20, 28, 9, 29, 25, 51, 29, 28, 48, 26, 31, 9, 32, 31, 60, 32, 28, 57, 32, 37, 9, 32, 34, 69, 56, 37, 66, 26, 40, 9, 32, 40, 78, 56, 40, 75, 38, 52, 9, 44, 37, 87, 50

Offset: 1

Altug Alkan, May 15 2018

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A005185, A240827, A244477.

[n le 9 select 10-n else Self(n-Self(n-3))+Self(n-Self(n-6)): n in [1..80]]; // Vincenzo Librandi, May 20 2018

f:= proc(n) option remember; procname(n-procname(n-3))+procname(n-procname(n-6)) end proc:
for i from 1 to 9 do f(i):= 10-i od:
map(f, [$1..100]); # Robert Israel, May 16 2018

Nest[Append[#, #[[1 + Length@ # - #[[-3]] ]] + #[[1 + Length@ # - #[[-6]] ]] ] &, Range[9, 1, -1], 77] (* Michael De Vlieger, May 20 2018 *)

q=vector(10^5); for(n=1, 9, q[n]=9-n+1);for(n=10, #q, q[n]=q[n-q[n-3]]+ q[n-q[n-6]]); q

A309650 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2; a(n) = a(n-a(n-2)) + a(n-a(n-3)) for n > 4.

Original entry on oeis.org

3, 1, 4, 2, 5, 3, 6, 9, 7, 5, 3, 11, 14, 7, 5, 8, 16, 19, 7, 5, 8, 21, 24, 12, 5, 8, 26, 29, 12, 5, 8, 31, 34, 12, 5, 13, 36, 39, 12, 5, 13, 41, 44, 12, 5, 13, 46, 49, 17, 5, 13, 51, 54, 17, 5, 13, 56, 59, 17, 5, 13, 61, 64, 17, 5, 18, 66, 69, 17, 5, 18, 71, 74, 17, 5, 18, 76, 79, 17, 5, 18, 81, 84, 22, 5

Offset: 1

Altug Alkan and Nathan Fox, Aug 11 2019

A well-defined quasi-periodic solution for recurrence (a(n) = a(n-a(n-2)) + a(n-a(n-3))).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.

Cf. A063892, A244477, A309492, A309567.

I:=[3,1,4,2]; [n le 4 select I[n] else  Self(n-Self(n-2)) + Self(n-Self(n-3)): n in [1..90]]; // Marius A. Burtea, Aug 11 2019

Nest[Append[#, #[[-#[[-2]] ]] + #[[-#[[-3]] ]]] &, {3, 1, 4, 2}, 81] (* Michael De Vlieger, May 08 2020 *)

q=vector(100); q[1]=3; q[2]=1; q[3]=4; q[4]=2; for(n=5, #q, q[n] = q[n-q[n-2]] + q[n-q[n-3]]); q

For k >= 1:
a(5*k)   = 5,
a(5*k+1) = 5*floor(sqrt(k)+1/2)-2,
a(5*k+2) = 5*k+1,
a(5*k+3) = 5*k+4,
a(5*k+4) = 5*floor(sqrt(k))+2.

A309967 a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 8, a(6) = 6, a(7) = a(8) = 4; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 8.

Original entry on oeis.org

1, 1, 2, 3, 8, 6, 4, 4, 9, 4, 8, 7, 9, 12, 6, 13, 7, 14, 17, 6, 18, 7, 19, 22, 6, 23, 7, 24, 27, 6, 28, 7, 29, 32, 6, 33, 7, 34, 37, 6, 38, 7, 39, 42, 6, 43, 7, 44, 47, 6, 48, 7, 49, 52, 6, 53, 7, 54, 57, 6, 58, 7, 59, 62, 6, 63, 7, 64, 67, 6, 68, 7, 69, 72, 6, 73, 7, 74, 77, 6, 78, 7

Offset: 1

Altug Alkan, Aug 25 2019

A quasilinear solution sequence for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A005185, A063882, A244477, A309567, A309636.

q=vector(100); q[1]=q[2]=1; q[3]=2; q[4]=3; q[5]=8; q[6]=6; q[7]=q[8]=4; for(n=9, #q, q[n]=q[n-q[n-1]]+q[n-q[n-4]]); q

Vec(x*(1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 4*x^5 + 2*x^6 + 3*x^8 - 12*x^9 - 3*x^10 +  3*x^12 - 3*x^13 + 6*x^14 + 3*x^15 - 3*x^16 + 2*x^18 - 2*x^19) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2) + O(x^40)) \\ Colin Barker, Aug 25 2019

For k > 2:
a(5*k-4) = 5*k-7,
a(5*k-3) = 7,
a(5*k-2) = 5*k-6,
a(5*k-1) = 5*k-3,
a(5*k) = 6.
From Colin Barker, Aug 25 2019: (Start)
G.f.: x*(1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 4*x^5 + 2*x^6 + 3*x^8 - 12*x^9 - 3*x^10 +  3*x^12 - 3*x^13 + 6*x^14 + 3*x^15 - 3*x^16 + 2*x^18 - 2*x^19) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = 2*a(n-5) - a(n-10) for n > 20.
(End)

A293947 Sequence P(n) arising in the analysis of the Hofstadter "brother" sequence A284644.

Original entry on oeis.org

1, 3, 8, 19, 41, 85, 173, 349, 701, 1405, 2800, 5576, 11128, 22221, 44342, 88422, 176507, 352062, 702831, 1403235, 2802382, 5598862, 11185734, 22353592, 44674558

Offset: 1

N. J. A. Sloane, Oct 26 2017

Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, Volume 2017, Article ID 2614163, 8 pages.

Cf. A004001, A005185, A244477, A284644.

A304622 a(n) = 11 - n for 1 <= n <= 10. Thereafter a(n) = a(n-a(n-2)) + a(n-a(n-4)).

Original entry on oeis.org

10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 6, 8, 10, 15, 8, 13, 6, 14, 16, 9, 14, 13, 6, 14, 22, 18, 20, 16, 6, 23, 28, 15, 26, 22, 6, 29, 34, 15, 32, 28, 6, 35, 40, 15, 38, 34, 6, 41, 46, 15, 44, 40, 6, 47, 52, 15, 50, 46, 6, 53, 58, 15, 56, 52, 6, 59, 64, 15, 62, 58, 6, 65, 70, 15, 68, 64, 6, 71

Offset: 1

Altug Alkan, May 15 2018

Cf. A087777, A240809, A244477, A304490.

f:= proc(n) option remember; procname(n-procname(n-2))+procname(n-procname(n-4)) end proc:
for i from 1 to 10 do f(i):= 11-i od:
map(f, [$1..100]); # Robert Israel, May 16 2018

q=vector(10^5); for(n=1, 10, q[n]=10-n+1); for(n=11, #q, q[n]=q[n-q[n-2]]+ q[n-q[n-4]]); q

a(6*k-3) = 6*(k-1)-4, a(6*k-2) = 6*(k-2)-2, a(6*k-1) = 6, a(6*k) = 6*(k-1)-1, a(6*k+1) = 6*k-2, a(6*k+2) = 15 for k > 4.

cp's OEIS Frontend

A304490 a(1) = a(2) = a(3) = 1, a(4) = 5, a(5) = 6, a(6) = 4; a(n) = a(n-a(n-2)) + a(n-a(n-4)) for n > 6.

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A304493 a(0) = 1; a(n) = a(n-a(floor(n/2))) + a(n-a(floor(n/4))).

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A304621 a(n) = 10 - n for 1 <= n <= 9. Thereafter a(n) = a(n-a(n-3)) + a(n-a(n-6)).

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A309650 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2; a(n) = a(n-a(n-2)) + a(n-a(n-3)) for n > 4.

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A309967 a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 8, a(6) = 6, a(7) = a(8) = 4; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 8.

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A293947 Sequence P(n) arising in the analysis of the Hofstadter "brother" sequence A284644.

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A304622 a(n) = 11 - n for 1 <= n <= 10. Thereafter a(n) = a(n-a(n-2)) + a(n-a(n-4)).

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Maple

PARI

Formula