A046700 -id:A046700 - cp's OEIS frontend

A063892 a(1) = a(2) = a(3) = 1, a(n) = a(n-a(n-2))+a(n-a(n-3)) for n>3.

1, 1, 1, 2, 4, 6, 5, 3, 3, 9, 6, 4, 7, 12, 9, 5, 7, 10, 16, 16, 10, 12, 12, 16, 14, 21, 13, 17, 8, 14, 24, 26, 19, 12, 8, 23, 22, 23, 12, 17, 18, 28, 35, 26, 16, 22, 34, 47, 22, 6, 10, 36, 69, 36

Offset: 1

Henry Bottomley, Aug 29 2001

Undefined for n>54 since a(53)=69>=55.

			a(7) = a(7-a(5))+a(7-a(4)) = a(7-4)+a(7-2) = a(3)+a(5) = 1+4 = 5.

Cf. A005185, A046700, A063882, A226222.

See A064714 for a variant which does not die.

#Q(r,s) with initial values 1,1,1,..., from N. J. A. Sloane, Apr 15 2014
r:=2; s:=3;
a:=proc(n) option remember; global r,s;
if n <= s then 1
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..54)];

A087777 a(1) = ... = a(4) = 1; a(n) = a(n - a(n-2)) + a(n - a(n-4)).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 7, 7, 5, 3, 8, 9, 11, 12, 9, 9, 13, 11, 9, 13, 16, 13, 19, 16, 11, 14, 16, 21, 22, 14, 14, 19, 17, 22, 27, 25, 16, 20, 28, 22, 22, 26, 25, 24, 32, 26, 22, 29, 29, 32, 35, 32, 27, 26, 34, 30, 33, 40, 25, 27, 46, 40, 33, 32, 28, 36, 50, 44, 31, 36, 38, 46, 53, 41, 29, 41

Offset: 1

Roger L. Bagula, Oct 05 2003

nonn

This is the sequence Q(2,4) in the Hofstadter-Huber classification.

It is not known if this sequence is defined for all positive n. Balamohan et al. comment that it shows "inscrutably wild behavior".

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
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. A005185 (Q(1,2)), A063882 (Q(1,4)), A046700.

[n le 4 select 1 else Self(n-Self(n-2))+Self(n-Self(n-4)): n in [1..80]]; // Vincenzo Librandi, Sep 10 2016

a := proc(n) option remember; if n<=4 then 1 else if n > a(n-2) and n > a(n-4) then RETURN(a(n-a(n-2))+a(n-a(n-4))); else ERROR(" died at n= ", n); fi; fi; end;

a[n_] := a[n] = If[n <= 4, 1, a[n - a[n - 2]] + a[n - a[n - 4]]];
Array[a, 80] (* Jean-François Alcover, Nov 24 2017 *)

Edited by N. J. A. Sloane, Nov 06 2007

A134680 a(n) = length (or lifetime) of the meta-Fibonacci sequence {f(1) = ... = f(n) = 1; 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, 164, 0, 60, 2354, 282, 1336, 100, 1254, 366, 419, 498, 483, 778, 1204, 292, 373, 845, 838, 1118, 2120, 815, 2616, 686, 1195, 745, 1112, 2132, 1588, 754, 1227, 1279, 1661, 716, 2275, 784, 2341, 1874, 1463, 1122, 2800, 1350, 1613, 2279, 1557, 1532

Offset: 1

T. D. Noe, Nov 06 2007

nonn

Such a sequence has finite length when the k-th term becomes greater than k.

The term a(2) = 0 is only conjectural - see A005185. a(4) = 0 is a theorem of Balamohan et al. (2007). - N. J. A. Sloane, Nov 07 2007, Apr 18 2014.

			a(1) = 6: the f-sequence is defined by f(1) = 1, f(n) = 2f(n-f(n-1)), which gives 1,2,2,4,2,8 but f(7) = 2f(-1) is undefined, so the length is 6.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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.
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 21500 terms.
Index entries for Hofstadter-type sequences

Cf. A005185, A046700, A063882, A132172, A134679 (sequences for n=2..6).
See A240810 for another version.
A diagonal of the triangle in A240813.

Table[Clear[a]; a[n_] := a[n] = If[n<=k, 1, a[n-a[n-1]]+a[n-a[n-k]]]; t={1}; n=2; While[n<10000 && a[n-1]

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

A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

Original entry on oeis.org

1, 2, 2, 7, 5, 1, 7, 2, 9, 3, 11, 3, 6, 4, 14, 5, 9, 16, 6, 15, 6, 10, 8, 21, 21, 21, 2, 28, 3, 30, 3, 6, 4, 33, 5, 9, 35, 6, 34, 6, 10, 8, 40, 40, 40, 2, 47, 3, 49, 3, 6, 4, 52, 5, 9, 54, 6, 53, 6, 10, 8, 59, 59, 59, 2, 66, 3, 68, 3, 6, 4, 71, 5, 9, 73, 6, 72, 6, 10, 8, 78, 78, 78, 2, 85, 3, 87, 3, 6, 4

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 07 2019

A well-defined solution sequence for recurrence a(n) = a(n-a(n-1)) + a(n-a(n-3)).

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

Cf. A046700, A244477, A296518, A309492, A309494, A309496.

I:=[1,2,2,7,5,1,7,2];[n le 8 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-3)): n in [1..90]]; // Marius A. Burtea, Aug 08 2019

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

Vec(x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2) + O(x^90)) \\ Colin Barker, Aug 11 2019

For k >= 1:
a(19*k-11) = 2,
a(19*k-10) = 19*k-10,
a(19*k-9) = 3,
a(19*k-8) = 19*k-8,
a(19*k-7) = 3,
a(19*k-6) = 6,
a(19*k-5) = 4,
a(19*k-4) = 19*k-5,
a(19*k-3) = 5,
a(19*k-2) = 9,
a(19*k-1) = 19*k-3,
a(19*k) = 6,
a(19*k+1) = 19*k-4,
a(19*k+2) = 6,
a(19*k+3) = 10,
a(19*k+4) = 8,
a(19*k+5) = a(19*k+6) = a(19*k+7) = 19*k+2.
From Colin Barker, Aug 08 2019: (Start)
G.f.: x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2).
a(n) = 2*a(n-19) - a(n-38) for n > 45.
(End)

cp's OEIS Frontend

A063892 a(1) = a(2) = a(3) = 1, a(n) = a(n-a(n-2))+a(n-a(n-3)) for n>3.

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A087777 a(1) = ... = a(4) = 1; a(n) = a(n - a(n-2)) + a(n - a(n-4)).

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A134680 a(n) = length (or lifetime) of the meta-Fibonacci sequence {f(1) = ... = f(n) = 1; 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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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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A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

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