A046699 -id:A046699 - cp's OEIS frontend

A241154 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=5.

1, 1, 1, 1, 1, 5, 5, 5, 5, 5, 9, 9, 9, 9, 13, 9, 13, 13, 17, 13, 17, 13, 17, 17, 21, 17, 21, 21, 21, 21, 25, 25, 25, 25, 25, 29, 29, 29, 29, 33, 29, 33, 33, 37, 33, 37, 33, 41, 37, 41, 37, 45, 37, 45, 41, 49, 41, 49, 41, 53, 45, 53, 45, 57, 45, 57, 49, 61, 49, 61, 49, 61, 53, 65, 53, 65, 57, 65, 57, 69, 61, 69, 61

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 Eq. (1.7).
Index entries for Hofstadter-type sequences

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:=5;
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)];

s = 0; k = 5; a[n_] := a[n] = If[n <= s + k, 1, Sum[a[n - i - s - a[n - i - 1]], {i, 0, k - 1}]]; Array[a, 100] (* Jean-François Alcover, Nov 10 2017 *)

A240835 a(n)=1 for n <= s+k; thereafter a(n) = Sum_{i=0..k-1} a(n-i-s-a(n-i-1)) where s=0, k=4.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 7, 7, 7, 10, 7, 10, 13, 10, 13, 13, 13, 16, 16, 16, 16, 16, 19, 19, 19, 22, 19, 22, 25, 22, 25, 25, 25, 28, 28, 28, 28, 31, 31, 31, 34, 31, 34, 37, 34, 37, 37, 34, 40, 40, 37, 43, 40, 40, 46, 43, 43, 46, 46, 46, 49, 49, 46, 52, 52, 49, 55, 52, 52, 58, 55, 55, 58, 55, 58, 61, 58, 61, 61, 61

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 Eq. (1.7) and Table 6.1
Index entries for Hofstadter-type sequences

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:=4;
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)];

A240835[n_]:=A240835[n]=If[n<=4,1,Sum[A240835[n-i-A240835[n-i-1]],{i,0,3}]];
Array[A240835,100] (* Paolo Xausa, Dec 06 2023 *)

A241155 a(n)=1 for n <= s+k; thereafter a(n) = Sum_{i=0..k-1} a(n-i-s-a(n-i-1)) where s=0, k=6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 16, 11, 16, 16, 16, 21, 16, 21, 16, 21, 26, 21, 26, 21, 26, 26, 26, 31, 26, 31, 31, 31, 31, 31, 36, 36, 36, 36, 36, 36, 36, 41, 41, 41, 41, 41, 46, 41, 46, 46, 46, 51, 46, 51, 46, 51, 56, 51, 56, 51, 56, 56, 56, 61, 56, 61, 61, 61, 61, 61, 66, 66, 66, 66, 66

Offset: 1

N. J. A. Sloane, Apr 16 2014

nonn

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 Eq. (1.7).
Index entries for Hofstadter-type sequences

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:=6;
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)];

A241155[n_]:=A241155[n]=If[n<=6,1,Sum[A241155[n-i-A241155[n-i-1]],{i,0,5}]];
Array[A241155,100] (* Paolo Xausa, Dec 06 2023 *)

A234016 Partial sums of the characteristic function of A055938.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 13, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 19, 20, 21, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 26, 26, 26, 27, 28, 29, 30, 31, 31, 31, 32, 32, 32, 33, 34

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Also: a(0) = a(1) = 0, and thereafter, a(n) = the largest k such that A055938(k) <= n.

Conjecture: partial sums of A308187 (i.e, A308187 is the characteristic function of A055938). - Sean A. Irvine, Jul 16 2022

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A046699, A055938, A079559, A234017, A233275, A308187.

from sympy import factorial
def a046699(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1)): s+=1
    return s
def a(n): return n - (a046699(n + 2) - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

If n < 2, a(n)=0, otherwise a(n) = a(n-1) + (1-A079559(n)).

a(n) = n - (A046699(n+2)-1) [With A046699's October 2012 starting offset].

A230413 a(0)=0 and from then on, the partial sums of A230412 summed from the term a(1) onward.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 10, 11, 11, 11, 11, 11, 11, 12, 13, 14, 14, 14, 15, 16, 17, 17, 17, 18, 19, 20, 20, 20, 21, 22, 23, 23, 23, 23, 23, 23, 24, 25, 26, 26, 26, 27, 28, 29, 29, 29, 30, 31, 32, 32, 32, 33, 34, 35, 35, 35, 35, 35, 35

Offset: 0

Antti Karttunen, Nov 02 2013

nonn

Alternatively, one less than the partial sums of A230412 summed from the beginning.
Each n occurs A230405(n) times.
Together with A230412 can be used to compute A230414, A230423 and A230424.

Antti Karttunen, Table of n, a(n) for n = 0..10079

Cf. also A230405, A230412, A230414.

This sequence relates to the factorial base representation (A007623) in a similar way as A046699 relates to the binary system.

a(0) = 0; and for n>=1, a(n) = A230413(n-1) + A230412(n).

a(A219650(n)) = n for all n.

A316628 a(1)=1, a(2)=2, a(3)=2, a(4)=3; a(n) = a(n-a(n-1))+a(n-1-a(n-2)-a(n-2-a(n-2))) for n > 4.

Original entry on oeis.org

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

Offset: 1

Nathan Fox, Jul 08 2018

nonn

This sequence increases slowly.
k occurs A035612(k) times.
Each Fibonacci number occurs more times than any number before it.

Nathan Fox, Table of n, a(n) for n = 1..10000
Nathan Fox, Trees, Fibonacci Numbers, and Nested Recurrences, Rutgers University Experimental Math Seminar, Mar 07, 2019

Cf. A000045, A005185, A035612, A046699.

a:=[1,2,2,3];; for n in [5..80] do a[n]:=a[n-a[n-1]]+a[n-1-a[n-2]-a[n-2-a[n-2]]]; od; a; # Muniru A Asiru, Jul 09 2018

I:=[1,2,2,3]; [n le 4 select I[n] else Self(n-Self(n-1))+Self(n-1-Self(n-2)-Self(n-2-Self(n-2))): n in [1..100]]; // Vincenzo Librandi, Jul 09 2018

A316628:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 1: elif n = 2 then 2: elif n = 3 then 2: elif n = 4 then 3: else A316628(n-A316628(n-1)) + A316628(n-1-A316628(n-2)-A316628(n-2-A316628(n-2))): fi: end:

a(n+1)-a(n)=1 or 0.

a(n)/n -> C=(sqrt(5)-1)/(sqrt(5)+1).

A324475 k appears t+1 times, where t is the number of trailing zeros in A324474(k).

Original entry on oeis.org

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

Offset: 1

Nathan Fox and N. J. A. Sloane, Mar 09 2019

Interesting because the recurrence is nested one layer deeper than the recurrences for A046699 and A316628.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Nathan Fox, Trees, Fibonacci Numbers, and Nested Recurrences, Rutgers University Experimental Math Seminar, Mar 07, 2019
Rémy Sigrist, PARI program for A324475

Cf. A324474.

A046699, A316628, A324473, A324477 have similar definitions.

PARI
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See Links section.
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For n>3, a(n) = a(n-a(n-1)) + a(n-1-a(n-2)-a(n-2-a(n-2))) + a(n-2-a(n-3)-a(n-3-a(n-3)) - a(n-3-a(n-3)-a(n-3-a(n-3)))). - Nathan Fox, Mar 09 2019 (This formula assumes that a(0) = 0. - Rémy Sigrist, Mar 14 2021)

Data corrected and more terms from Rémy Sigrist, Mar 14 2021

A324477 k appears t+1 times, where t = A364377(k) is the number of trailing zeros in the greedy Jacobsthal representation of k, A265747(k).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 9, 10, 10, 10, 11, 11, 11, 11, 12, 13, 14, 14, 15, 16, 16, 16, 17, 18, 19, 19, 20, 21, 21, 21, 21, 21, 22, 23, 24, 24, 25, 26, 26, 26, 27, 28, 29, 29, 30, 31, 31, 31, 32, 32, 32, 32, 33, 34, 35, 35, 36, 37, 37, 37, 38, 39, 40, 40

Offset: 1

Nathan Fox and N. J. A. Sloane, Mar 09 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nathan Fox, Trees, Fibonacci Numbers, and Nested Recurrences, Rutgers University Experimental Math Seminar, Mar 07, 2019.

Cf. A001045, A265747, A364377.

A046699, A316628, A324473 and A324475 have similar definitions.

Table[Table[k, {IntegerExponent[A265747[k], 10] + 1}], {k, 1, 40}] // Flatten (* Amiram Eldar, Jul 21 2023 using A265747[n] *)

More terms from Amiram Eldar, Jul 21 2023

A324473 k appears A278045(k)+1 times.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 1, 13, 13, 13, 13, 14, 15, 15, 16, 17, 17, 17, 18, 189, 19, 20, 20, 20, 20, 21, 22, 22, 23, 24, 24, 24, 24, 24, 24, 25, 26, 26, 27, 28, 28, 28, 29, 30, 30

Offset: 1

N. J. A. Sloane, Mar 07 2019

nonn

Each k appears one more times than the number of trailing zeros in the tribonacci representation of n (see A278038).

This is related to the tribonacci representation of n in the same way as A046699 (without its initial term) is related to the binary representation of n and as A316628 is to the Zeckendorf representation of n.

Nathan Fox, Trees, Fibonacci Numbers, and Nested Recurrences, Rutgers University Experimental Math Seminar, Mar 07, 2019

Cf. A046699, A278038, A278045, A316628.

A275363 a(1)=3, a(2)=6, a(3)=3; thereafter a(n) = a(n-a(n-1)) + a(n-1-a(n-2)).

Original entry on oeis.org

3, 6, 3, 3, 9, 6, 3, 12, 9, 3, 15, 12, 3, 18, 15, 3, 21, 18, 3, 24, 21, 3, 27, 24, 3, 30, 27, 3, 33, 30, 3, 36, 33, 3, 39, 36, 3, 42, 39, 3, 45, 42, 3, 48, 45, 3, 51, 48, 3, 54, 51, 3, 57, 54, 3, 60, 57, 3, 63, 60, 3, 66, 63, 3, 69, 66, 3, 72, 69

Offset: 1

Nathan Fox, Jul 24 2016

nonn

Same recurrence as in A046699 but with different starting values.

This sequence is quasilinear.

Nathan Fox, Table of n, a(n) for n = 1..1000
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A046699, A005185, A244477.

Flatten[Array[{3, 3*# + 6, 3*# + 3} &, 30, 0]] (* Paolo Xausa, Oct 23 2024 *)
LinearRecurrence[{0,0,2,0,0,-1},{3,6,3,3,9,6},80] (* Harvey P. Dale, Nov 27 2024 *)

a(3n) = 3n, a(3n+1) = 3, a(3n+2) = 3n+6.
a(n) = 2*a(n-3) - a(n-6) for n>6.
G.f.: -(3*x^4 +3*x^3 -3*x^2 -6*x-3)/((x-1)^2*(x^2+x+1)^2).

cp's OEIS Frontend

A241154 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=5.

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A240835 a(n)=1 for n <= s+k; thereafter a(n) = Sum_{i=0..k-1} a(n-i-s-a(n-i-1)) where s=0, k=4.

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A241155 a(n)=1 for n <= s+k; thereafter a(n) = Sum_{i=0..k-1} a(n-i-s-a(n-i-1)) where s=0, k=6.

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A234016 Partial sums of the characteristic function of A055938.

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A230413 a(0)=0 and from then on, the partial sums of A230412 summed from the term a(1) onward.

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A316628 a(1)=1, a(2)=2, a(3)=2, a(4)=3; a(n) = a(n-a(n-1))+a(n-1-a(n-2)-a(n-2-a(n-2))) for n > 4.

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A324475 k appears t+1 times, where t is the number of trailing zeros in A324474(k).

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A324477 k appears t+1 times, where t = A364377(k) is the number of trailing zeros in the greedy Jacobsthal representation of k, A265747(k).

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A324473 k appears A278045(k)+1 times.

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