A120503 -id:A120503 - cp's OEIS frontend

A120514 a(n) = min{j : A120503(j) = n}.

1, 2, 3, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 22, 23, 24, 27, 28, 29, 31, 32, 33, 35, 36, 37, 41, 42, 43, 45, 46, 47, 49, 50, 51, 54, 55, 56, 58, 59, 60, 62, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 81, 82, 83, 85, 86, 87, 89, 90, 91, 94, 95, 96, 98

Offset: 1

Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca), Jun 20 2006

nonn

Benoit Cloitre, 3^13 steps of the walk using step: turn 90° right if a(n)is even and turn 90° left if a(n) is odd
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences, J. Integer Seq., Vol. 12. [This is a later version than that in the GenMetaFib.html link]
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences

Cf. A120503, A120525, A005187.

p := proc(n)
if n=1 then return 1; end if;
for j from p(n-1)+1 to infinity do
if A120503(j) = n then return j; fi; od;
end proc;

G.f.: P(z) = z / (1-z) * (1 + Sum_{m>=0} z^(m^3)/(1 - z^(m^3))).

a(1) = 1; a(n) = n-1 + a(floor((n-1)/3)+1). - Gleb Ivanov, Jan 10 2022

A120525 First differences of successive generalized meta-Fibonacci numbers A120503.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 1

Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca), Jun 20 2006

nonn

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences, J. Integer Seq., Vol. 12. [This is a later version than that in the GenMetaFib.html link]
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences

Cf. A120503, A120514.

d := n -> if n=1 then 1 else A120503(n)-A120503(n-1) fi;

d(n) = 0 if node n is an inner node, or 1 if node n is a leaf.
g.f.: z (1 + z^1 ( (1 - z^(2 * [1])) / (1 - z^[1]) + z^3 * (1 - z^(3 * [i]))/(1 - z^[1]) ( (1 - z^(2 * [2])) / (1 - z^[2]) + z^9 * (1 - z^(3 * [2]))/(1 - z^[2]) (..., where [i] = (3^i - 1) / 2.
g.f.: D(z) = z * prod((1 - z^(3 * [i])) / (1 - z^[i])), i=1..infinity), where [i] = (3^i - 1) / 2.

A240830 a(n)=1 for n <= s+k; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13, 19, 13, 19, 19, 19, 19, 25, 19, 25, 19, 25, 25, 31, 25, 31, 25, 31, 25, 31, 31, 37, 31, 37, 31, 37, 37, 37, 37, 43, 37, 43, 43, 43, 43, 43, 43, 49, 49, 49, 49, 49, 49, 49, 55, 55, 55, 55, 55, 55, 61, 55, 61, 61, 61, 61, 67, 61, 67, 61, 67, 67, 73

Offset: 1

N. J. A. Sloane, Apr 16 2014

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See Table 2.1.
Index entries for Hofstadter-type sequences

Same recurrence as A240828, A120503 and A046702.
See also A240831, A240832.
Callaghan et al. (2005)'s sequences T_{0,k}(n) for k=1 through 7 are A000012, A046699, A046702, A240835, A241154, A241155, A240830.

#T_s,k(n) from Callaghan et al. Eq. (1.7).
s:=0; k:=7;
a:=proc(n) option remember; global s,k;
if n <= s+k then 1
else
    add(a(n-i-s-a(n-i-1)),i=0..k-1);
fi; end;
t1:=[seq(a(n),n=1..100)];

A240830[n_]:=A240830[n]=If[n<=7,1,Sum[A240830[n-i-A240830[n-i-1]],{i,0,6}]];
Array[A240830,100] (* Paolo Xausa, Dec 06 2023 *)

A007844 Least positive integer k for which 3^n divides k!.

Original entry on oeis.org

1, 3, 6, 9, 9, 12, 15, 18, 18, 21, 24, 27, 27, 27, 30, 33, 36, 36, 39, 42, 45, 45, 48, 51, 54, 54, 54, 57, 60, 63, 63, 66, 69, 72, 72, 75, 78, 81, 81, 81, 81, 84, 87, 90, 90, 93, 96, 99, 99, 102, 105, 108, 108, 108, 111, 114, 117, 117, 120, 123, 126, 126, 129, 132, 135, 135, 135

Offset: 0

Bruce Dearden and Jerry Metzger, R. Muller

nonn

It appears than for n>0, a(n) is divisible by 3, and that the resulting sequence a(n)/3 is A120503 (checked up to n=1000). - Michel Marcus, Aug 19 2013. [This is true: see A007843 for the idea of the proof. - M. F. Hasler, Dec 27 2019]

Also least positive integer k for which 6^n divides k!. - Michel Marcus, Aug 20 2013

H. Ibstedt, Smarandache Primitive Numbers, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 216-229.

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Smarandache, Only Problems, Not Solutions!

Cf. A007843 (analog for 2), A007845 (analog for 5).

Cf. A120503 (Meta-Fibonacci, k = 3).

Array[Block[{k = 1}, While[Mod[k!, 3^#] != 0, k++]; k] &, 67, 0] (* Michael De Vlieger, Dec 29 2019 *)

a(n) = {k = 1; while (valuation(k!, 3) < n, k++); k;} \\ Michel Marcus, Aug 19 2013

apply( A007844(n)={my(s=sumdigits(n*=2,3)\2); n-=n%3; while(s>0, s-=valuation(n+=3,3)); n+!n}, [0..99]) \\ M. F. Hasler, Dec 27 2019

a(n) = 3*A120503(n) for n > 0, cf. A007843. - M. F. Hasler, Dec 27 2019

A228297 Generalized meta-Fibonacci sequence a(n) with parameters s=0 and k=5.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 50, 50, 51, 52, 53

Offset: 1

Michel Marcus, Aug 20 2013

nonn

Each integer n appears x+1 times where x is the greatest power of 5 in the factorization of n!. - Gerald Hillier, Feb 08 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences, J. Integer Seq., Vol. 12, 2009, Article 09.4.3.

Cf. A120503, A120507, A228298.

Array[ConstantArray[#, IntegerExponent[#, 5] + 1] &, 53] // Flatten (* Michael De Vlieger, Feb 08 2020 *)

a(n)= {local(A); if(n<=5, max(0, n), A=vector(n, i, i); for(k=6, n, A[k]= A[k-A[k-1]] + A[k-1-A[k-2]] + A[k-2-A[k-3]] + A[k-3-A[k-4]] + A[k-4-A[k-5]];); A[n];);}
(HP 49G calculator)
« DUPDUP 5 IQUOT -
  WHILE DUP 0 OVER
    DO 5 IQUOT DUP
ROT + SWAP DUP NOT
    UNTIL
    END DROP +
PICK3 <
  REPEAT 1 +
  END NIP
» Gerald Hillier, Sep 19 2017

a(n) = A007845(n)/5. - M. F. Hasler, Dec 27 2019

A228298 Generalized meta-Fibonacci sequence a(n) with parameters s=0 and k=7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 49, 49, 50, 51, 52, 53, 54, 55, 56, 56

Offset: 1

Michel Marcus, Aug 20 2013

nonn

It appears than a(n) = A020646(n)/7 for n>0 (verified up to n=700).

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences, J. Integer Seq., Vol. 12.

Cf. A120503, A120507, A228297.

a(n)= {local(A); if(n<=7, max(0, n), A=vector(n, i, i); for(k=8, n, A[k]= A[k-A[k-1]] + A[k-1-A[k-2]] + A[k-2-A[k-3]] + A[k-3-A[k-4]] + A[k-4-A[k-5]] + A[k-5-A[k-6]] + A[k-6-A[k-7]];); A[n];);}

A240829 a(1)=-1, a(2)=0, a(3)=1; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=3.

Original entry on oeis.org

-1, 0, 1, 3, 2, 4, 4, 7, 4, 7, 7, 9, 8, 9, 11, 10, 10, 13, 15, 13, 13, 13, 18, 15, 18, 18, 18, 18, 18, 23, 23, 20, 19, 23, 28, 27, 23, 25, 27, 28, 25, 26, 28, 30, 31, 32, 33, 33, 32, 34, 33, 38, 36, 39, 34, 36, 36, 39, 39, 39, 39, 44, 46, 46, 43, 46, 46, 44, 44, 49, 49, 49, 46, 51, 48, 51, 51, 54, 54, 54, 54, 54

Offset: 1

N. J. A. Sloane, Apr 16 2014

Callaghan, Joseph, John J. Chew III, and Stephen M. Tanny. "On the behavior of a family of meta-Fibonacci sequences." SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See Fig. 1.7.

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

Same recurrence as A240828, A120503 and A046702.

#T_s,k(n) from Callaghan et al. Eq. (1.6).
s:=0; k:=3;
a:=proc(n) option remember; global s,k;
if n <= 3 then n-2
else
    add(a(n-i-s-a(n-i-1)),i=0..k-1);
fi; end;
t1:=[seq(a(n),n=1..100)];

cp's OEIS Frontend

A120514 a(n) = min{j : A120503(j) = n}.

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A120525 First differences of successive generalized meta-Fibonacci numbers A120503.

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A240830 a(n)=1 for n <= s+k; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=7.

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A007844 Least positive integer k for which 3^n divides k!.

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A228297 Generalized meta-Fibonacci sequence a(n) with parameters s=0 and k=5.

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Mathematica

PARI

Formula

A228298 Generalized meta-Fibonacci sequence a(n) with parameters s=0 and k=7.

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PARI

A240829 a(1)=-1, a(2)=0, a(3)=1; thereafter a(n) = Sum(a(n-i-s-a(n-i-1)),i=0..k-1) where s=0, k=3.

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Maple