A004001 -id:A004001 - cp's OEIS frontend

A285764 A relative of the Hofstadter-Conway sequence A004001.

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

Offset: 1

Nathan Fox, Apr 25 2017

nonn

a(n) is the solution to the recurrence relation a(n) = a(a(n-3)) + a(n-a(n-3)), with the initial conditions: a(1) = 1, a(2) = 2, a(3) = a(4) = a(5) = a(6) = 3, a(7) = 4, a(8) = 5, a(9) = 6.
The sequence a(n) is monotonic, with successive terms increasing by 0 or 1. So the sequence hits every positive integer.
This sequence can be obtained from the Hofstadter-Conway sequence A004001 using a construction of Isgur et al.

Nathan Fox, Table of n, a(n) for n = 1..10000
A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505

Cf. A004001, A285763.

A285764:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 1: elif n = 2 then 2: elif n = 3 then 3: elif n = 4 then 3: elif n = 5 then 3: elif n = 6 then 3: elif n = 7 then 4: elif n = 8 then 5: elif n = 9 then 6: else A285764(A285764(n-3)) + A285764(n-A285764(n-3)): fi: end:

A302780 Restricted growth sequence transform of 4-tuple [H(H(n-1)), H(n-H(n-1)), Q(n-Q(n-1)), Q(n-Q(n-2))] where H = A004001 and Q = A005185.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 47, 50, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 79, 80, 80

Offset: 1

Antti Karttunen, Apr 27 2018

nonn

Restricted growth sequence transform of A286560: a filter sequence which includes both the summands of A004001 and the summands of A005185.
For all i, j: a(i) = a(j) => b(i) = b(j), where b is a sequence like A087740, A284019, A286569 or A302779.
For n > 1000 the duplicates get rare. In range [1000, 65536] there are only three cases: a(1353) = a(1354) = 1319, a(39361) = a(39362) = 39326, and a(46695) = a(46696) = 46659.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for Hofstadter-type sequences

Cf. A004001, A005185, A286560.

Cf. also A087740, A284019, A286541, A286569, A302779.

up_to = 65537;
first_n_of_A004001(n) = { my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); (v); }; \\ Charles R Greathouse IV, Feb 26 2017
v004001 = first_n_of_A004001(up_to);
A004001(n) = v004001[n];
first_n_of_A005185(n) = { my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[k-v[k-1]]+v[k-v[k-2]]); (v); }; \\
v005185 = first_n_of_A005185(up_to);
A005185(n) = v005185[n];
Aux302780(n) = if(n<3,0,[A004001(A004001(n-1)), A004001(n-A004001(n-1)), A005185(n-A005185(n-1)), A005185(n-A005185(n-2))]);
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302780(n))),"b302780.txt");

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)

A089990 a(n) = n step nested walk based on (A004001[n]+ (-1)^lastsum).

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, Jan 14 2004

nonn

A walk based on A004001.

Cf. A004001, A089991.

Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 digits=20 b=Table[Conway[n], {n, 1, digits}] ChaosWalk[n_Integer?Positive] := NestList[(b[[n]]+(-1)^#)&, 0, n] ChaosWalk[1]=1 a=Flatten[Table[ChaosWalk[n], {n, 1, digits}]]

A089991 Maximum lengths of chaotic walks based on the A004001 sequence.

Original entry on oeis.org

0, 0, 6, 8, 0, 24, 28, 32, 0, 60, 0, 0, 104, 112, 120, 128, 0, 180, 0, 240, 252, 0, 322, 336, 0, 0, 0, 448, 464, 480, 496, 512, 0, 612, 0, 720, 0, 0, 858, 0, 984, 1008, 0, 1144, 1170, 0, 0, 0, 1372, 0, 0, 1560, 1590, 1620, 0, 0, 0, 0, 1888, 1920, 1952, 1984, 2016, 2048, 0

Offset: 1

Roger L. Bagula, Jan 14 2004

nonn

Cf. A004001, A089990.

Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 digits=200 b=Table[(-1)^Conway[n]*Conway[n], {n, 1, digits}] ChaosWalk[n_Integer?Positive] := NestList[(#+b[[n]])&, 0, n] ChaosWalk[1]=0 a=Table[Max[ChaosWalk[n]], {n, 1, digits}]

a(n) = MaximumLength[ nested n walk based on A004001[n]*(-1)^A004001[n]+lastsum]

A095767 a(n) = valuation(A004001(n),2).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 2, 0, 1, 0, 0, 3, 3, 3, 3, 0, 1, 0, 2, 2, 0, 1, 1, 0, 0, 0, 4, 4, 4, 4, 4, 0, 1, 0, 2, 0, 0, 1, 0, 3, 3, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 5, 5, 5, 5, 5, 5, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 4, 4, 4, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 0, 3, 3, 0, 0, 0, 1, 1, 1

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

Cf. A095768.

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; f[n_] := Length[ NestWhileList[ #/2 &, n, IntegerQ[ # ] &]] - 2; Table[ f[ a[n]], {n, 105}] (* Robert G. Wilson v, Jun 11 2004 *)

Partial formula: a(2^(n+1) - n + i) = n for 0<=i<=n.

A095900 a(n) = A004001(10^n).

Original entry on oeis.org

1, 6, 57, 510, 5373, 53505, 510403, 5247173, 52736107, 511172800, 5189628970, 52334438874, 511861449132, 5150236044255, 52074775905991, 512279427101305, 5118687220533539, 51879201305335167, 512519244788358058

Offset: 0

Robert G. Wilson v, Jun 11 2004

nonn

a(n)/n -> 10^n/(2*n). [Corrected by Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007]

Herman Jamke, Table of n, a(n) for n = 0..30
C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20. See comments on formula due to G. Phillips on page 18.

Cf. A004001.

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Do[ a[n], {n, 1000000}]; Table[ a[10^n], {n, 0, 6}]

print1("1, ");for(k=1,30,n=10^k;row=floor(log(n)/log(2));col=1;s=0;a=0;while(s1,smd=binomial(row,j-2),smd=2^row);if((s+smd)>n,col=j-1;row=row-1;break,s+=smd;if(j>1,a+=binomial(row-1,j-2),a+=2^(row-1)))));print1(a", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

2 more terms from Ryan Propper, Jan 05 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

A116592 a(0)=1; a(n) = b(n+2) + b(n), where b(n) = A004001(n) is the Hofstadter-Conway sequence defined by b(1) = b(2) = 1, b(n) = b(b(n-1)) + b(n-b(n-1)) for n>2.

Original entry on oeis.org

1, 3, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15, 15, 16, 16, 17, 18, 20, 22, 23, 25, 26, 27, 29, 29, 30, 31, 31, 32, 32, 32, 33, 34, 36, 38, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 53, 54, 55, 56, 57, 59, 59, 60, 61, 61, 62, 62, 63, 63, 64, 64, 64, 64, 65, 66, 68, 70, 72, 74, 75, 77, 78

Offset: 0

Roger L. Bagula, Mar 27 2006

nonn

A similar definition applied to the Fibonacci sequence (A000045) leads to the Lucas sequence (A000032).

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000032, A005185, A005229, A004001.

b:=proc(n) option remember; if n<=2 then 1 else b(b(n-1))+b(n-b(n-1)): fi: end: seq(b(n),n=1..71): a:=proc(n) if n=0 then 1 else b(n+2)+b(n) fi end: seq(a(n),n=0..71);

Cw[0] = 0; Cw[1] = Cw[2] = 1; Cw[n_Integer?Positive] := Cw[n] = Cw[Cw[n - 1]] + Cw[n - Cw[n - 1]]; L[0] = 1; L[n_] := L[n] = Cw[n - 1] + Cw[n + 1]; Table[L[n], {n, 1, 200}]

a(n) = A004001(n+2) + A004001(n) for n>=1.

Edited by N. J. A. Sloane, Apr 15 2006

A121459 Let f(n) = A004001(n)^2 - A005185(n)^2. Then a(n) = f(abs(f(n-1))) + f(abs(n - f(n-1))).

Original entry on oeis.org

0, -3, -14, -18, -7, -9, -47, -51, 0, -15, -15, -48, -17, -36, 57, -151, 0, -63, 0, -11, 0, 25, 26, 368, 29, -5, -96, -33, 0, -144, 2275, -466, -180, 433, 472, 0, -43, 316, 0, 0, 47, -302, 49, 152, 1122, 945, 1273, 10170, 589, 1310, 121, 54, 3117, 0, 177, 2141, -1280, -5, 310, 0

Offset: 1

Roger L. Bagula, Sep 06 2006

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A004001, A005185.

HConway[n_]:= HConway[n]= If[n<3, 1, HConway[HConway[n-1]] + HConway[n-HConway[n -1]]];
Hofstadter[n_]:= Hofstadter[n]= If[n<3, 1, Hofstadter[n -Hofstadter[n-1]] + Hofstadter[n -Hofstadter[n-2]]];
f[n_]:= f[n]= HConway[n]^2 - Hofstadter[n]^2;
Table[f[Abs[f[n-1]]] + f[Abs[n -f[n-1]]], {n,60}] (* corrected by G. C. Greubel, Feb 12 2020 *)

Edited by N. J. A. Sloane, Oct 01 2006

Terms a(31) and beyond corrected by G. C. Greubel, Feb 12 2020

A135687 a(n) = a(n-1) - A004001(n)*a(n-2), a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, -1, -3, 0, 12, 12, -36, -96, 120, 792, -48, -6384, -6000, 45072, 93072, -312576, -1243296, 2195040, 17114592, -9225888, -231715584, -102553152, 3141465024, 4679762304, -42442213056, -112638647616, 566436761280, 2368655123136, -6694333057344, -44592815027520

Offset: 1

Roger L. Bagula, Feb 19 2008

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A004001, A135688.

HC[n_]:= HC[n]= If[n<3, Fibonacci[n], HC[HC[n-1]] +HC[n -HC[n-1]]]; (*A004001*)
a[n_]:= a[n]= If[n<3, 1, a[n-1] - HC[n]*a[n-2]];
Table[a[n], {n, 40}]

@cached_function
def HC(n): # HC = A004001
    if (n<3): return fibonacci(n)
    else: return HC(HC(n-1)) +HC(n -HC(n-1))
@CachedFunction
def a(n): # A135688
    if (n<3): return 1
    else: return a(n-1) - HC(n)*a(n-2)
[a(n) for n in (1..40)] # G. C. Greubel, Nov 25 2021

a(n) = a(n-1) - A004001(n)*a(n-2), with a(1) = a(2) = 1.

cp's OEIS Frontend

A285764 A relative of the Hofstadter-Conway sequence A004001.

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A302780 Restricted growth sequence transform of 4-tuple [H(H(n-1)), H(n-H(n-1)), Q(n-Q(n-1)), Q(n-Q(n-2))] where H = A004001 and Q = A005185.

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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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A089990 a(n) = n step nested walk based on (A004001[n]+ (-1)^lastsum).

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A089991 Maximum lengths of chaotic walks based on the A004001 sequence.

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Formula

A095767 a(n) = valuation(A004001(n),2).

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Mathematica

Formula

A095900 a(n) = A004001(10^n).

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Extensions

A116592 a(0)=1; a(n) = b(n+2) + b(n), where b(n) = A004001(n) is the Hofstadter-Conway sequence defined by b(1) = b(2) = 1, b(n) = b(b(n-1)) + b(n-b(n-1)) for n>2.

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A121459 Let f(n) = A004001(n)^2 - A005185(n)^2. Then a(n) = f(abs(f(n-1))) + f(abs(n - f(n-1))).

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