A240821 -id:A240821 - cp's OEIS frontend

A240809 a(n) = n for 1 <= n <= 4; thereafter a(n) = a(n - a(n-2)) + a(n - a(n-4)).

1, 2, 3, 4, 6, 6, 5, 6, 7, 8, 10, 10, 9, 10, 12, 12, 12, 12, 10, 12, 17, 16, 15, 16, 14, 16, 19, 20, 20, 18, 20, 20, 18, 22, 24, 22, 19, 24, 24, 24, 28, 24, 22, 24, 27, 32, 26, 28, 31, 28, 26, 32, 35, 32, 32, 32, 30, 32, 39, 40, 36, 34, 35, 34, 38, 40, 40, 42, 40, 38, 40, 40, 36, 44, 48, 42, 48, 44, 40, 46, 46, 46, 41, 48, 48, 48, 56, 48, 46, 48, 51, 48

Offset: 1

N. J. A. Sloane, Apr 15 2014, Apr 17 2014

nonn

Conjectured to be infinite.

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.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
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, Plot of first 100000 terms
Index entries for Hofstadter-type sequences

Cf. A240821.

I:=[1, 2, 3, 4]; [n le 4 select I[n] else Self(n-Self(n-2))+Self(n-Self(n-4)): n in [1..100]]; // Vincenzo Librandi, Apr 16 2014

# Q(r,s) with initial values 1,2,3,4,...
r:=2; s:=4;
a:=proc(n) option remember; global r,s;
if n <= s then n
else
    if (a(n-r) <= n) and (a(n-s) <= n) then
    a(n-a(n-r))+a(n-a(n-s));
    else lprint("died with n =",n); return (-1);
    fi;
fi; end;
[seq(a(n),n=1..100)];

A240825 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = index of first nonexisting term of the meta-Fibonacci sequence {f(i)=i for i <= n; thereafter f(i)=f(i-f(i-k))+f(i-f(i-n))} if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

Original entry on oeis.org

7, 0, 14, 163, 30, 21, 0, 0, 72, 28, 57, 30, 35, 36, 29, 2350, 25, 0, 29, 55, 42, 277, 51, 47, 45, 35, 56, 41, 1301, 0, 35, 0, 38, 69, 90

Offset: 1

N. J. A. Sloane, Apr 15 2014

The zero entries are only conjectural. More precisely, Hofstadter conjectures that T(n,k) = 0 (i.e., the sequence is immortal) iff n = 2k or n = 4k.

Apart from the zero entries, equals A240821 + 1.

			Triangle begins:
     7;
     0, 14;
   163, 30, 21;
     0,  0, 72, 28;
    57, 30, 35, 36, 29;
  2350, 25,  0, 29, 55, 42;
   277, 51, 47, 45, 35, 56, 41;
  1301,  0, 35,  0, 38, 69, 90, ...
  ...

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.

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.
Index entries for Hofstadter-type sequences

Diagonals give A240822, A240823, A240824.

See A240821 for another version.

A240827 a(n) = n for 1<=n<=6; thereafter a(n) = a(n-a(n-3))+a(n-a(n-6)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 9, 9, 7, 8, 9, 10, 11, 12, 15, 15, 15, 13, 14, 15, 18, 18, 18, 18, 18, 18, 14, 16, 18, 25, 26, 24, 23, 22, 24, 20, 24, 24, 29, 28, 30, 29, 29, 27, 30, 28, 30, 27, 33, 33, 36, 32, 33, 27, 36, 36, 43, 36, 36, 38, 36, 36, 33, 32, 36, 39, 50, 48, 45, 39, 42, 37, 40, 42, 49, 44, 48, 48, 53, 48, 47, 42, 48, 44, 53, 48, 57, 52, 60

Offset: 1

N. J. A. Sloane, Apr 15 2014

Conjectured to be infinite.

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.

N. J. A. Sloane, Table of n, a(n) for n = 1..50000
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.
Index entries for Hofstadter-type sequences

Cf. A240821.

I:=[1,2,3,4,5,6]; [n le 6 select I[n] else Self(n-Self(n-3))+Self(n-Self(n-6)): n in [1..100]]; // Vincenzo Librandi, Apr 16 2014

#Q(r,s) with initial values 1,2,3,4,...
r:=3; s:=6;
a:=proc(n) option remember; global r,s;
if n <= s then n
else
if (a(n-r) <= n) and (a(n-s) <= n) then
a(n-a(n-r))+a(n-a(n-s));
else lprint("died with n =",n); return (-1);
fi;
fi; end;
t2:=[seq(a(n),n=1..100)];

{a[1]=1,a[2]=2,a[3]=3,a[4]=4,a[5]=5,a[6]=6,a[n_]:=a[n]=a[n-a[n-3]]+ a[n-a[n-6]]};Table[a[x],{x,100}] (* Harvey P. Dale, Nov 18 2021 *)

A240818 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k)=f(k-f(k-1))+f(k-f(k-n)) if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

Original entry on oeis.org

6, 0, 162, 0, 56, 2349, 276, 1300, 84, 1245, 356, 408, 486, 470, 764, 1172, 258, 356, 805, 819, 1078, 2099, 470, 2593, 662, 1170, 665, 1085, 2104, 1417, 724, 1196, 1247, 1628, 648, 2240, 712, 2304, 1836, 1424, 1082, 2759, 1264, 1570, 2235, 1512, 1442, 2447

Offset: 1

N. J. A. Sloane, Apr 15 2014

nonn

The terms a(2) = 0 and a(4) = 0 are only conjectural.

This sequence is very similar to A134680.

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 = 1..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.
Index entries for Hofstadter-type sequences

The sequences for n=2,3,4 are A005185 and (essentially) A046700, A063882.
See A240822 for another version.
A diagonal of the triangle in A240821.
Cf. A134680.

More terms from Lars Blomberg, Oct 24 2014

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.

Original entry on oeis.org

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

A240820 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k) = f(k-f(k-3)) + f(k-f(k-n)) if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

Original entry on oeis.org

20, 71, 34, 0, 46, 34, 36, 39, 90, 0, 142, 70, 91, 94, 255, 2004, 306, 525, 259, 454, 304, 1866, 316, 198, 254, 297, 415, 3315, 348, 406, 397, 420, 903, 1226, 408, 589, 1294, 535, 490, 958, 1343, 477, 492, 915, 1378, 1723, 797, 1869, 745, 696, 863, 1070, 560

Offset: 3

N. J. A. Sloane, Apr 15 2014

nonn

The term a(6) = 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 = 3..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.
Index entries for Hofstadter-type sequences

See A240827 for the sequence for n=6.
See A240823 for another version.
A diagonal of the triangle in A240821.

More terms from Lars Blomberg, Oct 24 2014

cp's OEIS Frontend

A240809 a(n) = n for 1 <= n <= 4; thereafter a(n) = a(n - a(n-2)) + a(n - a(n-4)).

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A240825 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = index of first nonexisting term of the meta-Fibonacci sequence {f(i)=i for i <= n; thereafter f(i)=f(i-f(i-k))+f(i-f(i-n))} if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

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A240827 a(n) = n for 1<=n<=6; thereafter a(n) = a(n-a(n-3))+a(n-a(n-6)).

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Magma

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Mathematica

A240818 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k)=f(k-f(k-1))+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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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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Extensions

A240820 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k) = f(k-f(k-3)) + 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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