A240809 -id:A240809 - cp's OEIS frontend

A240819 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k)=f(k-f(k-2))+f(k-f(k-n)) if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

13, 29, 0, 29, 24, 50, 0, 332, 56, 848, 2936, 140, 370, 605, 1514, 532, 169, 360, 1784, 514, 713, 279, 817, 945, 973, 949, 932, 444, 1529, 420, 2345, 628, 517, 913, 713, 738, 1611, 1066, 1639, 727, 1256, 1140, 1336, 718, 941, 907, 2272, 606, 1152, 2091, 2341

Offset: 2

N. J. A. Sloane, Apr 15 2014

nonn

The term a(4) = 0 is only conjectural.

D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014.

Lars Blomberg, Table of n, a(n) for n = 2..10000, "infinity" = 10^8.
D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; Part 1, Part 2.
D. R. Hofstadter, Graph of first 30000 terms
Index entries for Hofstadter-type sequences

See A240809 for the sequence for n=4.
See A240823 for another version.
A diagonal of the triangle in A240821.

More terms from Lars Blomberg, Oct 24 2014

A283426 a(1) = a(2) = a(3) = a(4) = 1; a(n) = a(n-a(n-2)-1) + a(n-a(n-4)-2) for n > 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 4, 6, 8, 8, 6, 8, 8, 8, 8, 8, 10, 14, 14, 12, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 18, 24, 24, 22, 22, 16, 18, 24, 26, 28, 24, 22, 26, 24, 24, 28, 26, 26, 28, 26, 28, 32, 30, 30, 32, 30, 32, 32, 32, 32, 32, 32, 32, 34, 42, 42

Offset: 1

Altug Alkan, May 15 2017

For n < 10^5, values of a(n) such that a(n) = a(n + k) for all k = 1..6 are 2^4, 2^5, .., 2^14, 2^15.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Line graph of A283426

Cf. A005185, A240809.

a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = a[n - a[n - 2] - 1] + a[n - a[n - 4] - 2]; Array[a, 74] (* Michael De Vlieger, May 15 2017 *)

q=vector(10000); q[1]=q[2]=q[3]=q[4]=1; for(n=5, #q, q[n]=q[n-1-q[n-2]]+q[n-2-q[n-4]]); vector(10000, n, q[n])

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.

Original entry on oeis.org

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)

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

A240819 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k)=f(k-f(k-2))+f(k-f(k-n)) if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

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A283426 a(1) = a(2) = a(3) = a(4) = 1; a(n) = a(n-a(n-2)-1) + a(n-a(n-4)-2) for n > 4.

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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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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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