A080457 -id:A080457 - cp's OEIS frontend

A080578 a(1)=1; for n > 1, a(n) = a(n-1) + 1 if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.

1, 4, 7, 8, 11, 14, 15, 16, 19, 22, 23, 26, 29, 30, 31, 32, 35, 38, 39, 42, 45, 46, 47, 50, 53, 54, 57, 60, 61, 62, 63, 64, 67, 70, 71, 74, 77, 78, 79, 82, 85, 86, 89, 92, 93, 94, 95, 98, 101, 102, 105, 108, 109, 110, 113, 116, 117, 120, 123, 124, 125, 126

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Mar 23 2003

nonn

More generally for fixed r, there is a nice connection between the sequence a(1)=1, a(n) = a(n-1) + 1 if n is in the sequence, a(n) = a(n-1) + r + 1 otherwise and the so-called metafibonacci sequences. Indeed, (a(n)-n)/r is a generalized metafibonacci sequence of order r as defined in Ruskey's recent paper (reference given at A046699). - Benoit Cloitre, Feb 04 2007

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,1,3)-hiccup sequence. - Michael De Vlieger, Jul 29 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 3, 5, 11.

Cf. A080455, A080456, A080457, A080458, A080036, A080037, A080468.

a080578 n = a080578_list !! (n-1)
a080578_list = 1 : f 2 [1] where
   f x zs@(z:_) = y : f (x + 1) (y : zs) where
     y = if x `elem` zs then z + 1 else z + 3
-- Reinhard Zumkeller, Sep 26 2014

l={1}; a=1; For[n=2, n<=100, If[MemberQ[l, n], a=a+1, a=a+3]; AppendTo[l, a]; n++]; l (* Indranil Ghosh, Apr 07 2017 *)

a(n)=if(n<2,1,a(n+1-2^floor(log(n)/log(2)))+2*2^floor(log(n)/log(2))-1) \\ Benoit Cloitre, Feb 04 2007

l=[1]
a=1
for n in range(2, 101):
    a += 3 if n not in l else 1
    l.append(a)
print(l) # Indranil Ghosh, Apr 07 2017

a(n) = 2n + O(1); a(2^n) = 2^(n+1). - Benoit Cloitre, Oct 12 2003
a(1) = 1, for n >= 2 a(n) = a(n + 1 - 2^floor(log(n)/log(2))) + 2*2^floor(log(n)/log(2)) - 1; (a(n) - n)/2 = A046699(n) for n >= 2. - Benoit Cloitre, Feb 04 2007
a(n) = A055938(n-1) + 2 (conjectured). - Ralf Stephan, Dec 27 2013

A080458 a(1)=4; for n>1, a(n)=a(n-1) if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

Original entry on oeis.org

4, 8, 12, 12, 16, 20, 24, 24, 28, 32, 36, 36, 40, 44, 48, 48, 52, 56, 60, 60, 64, 68, 72, 72, 76, 80, 84, 84, 88, 92, 96, 96, 100, 104, 108, 108, 112, 116, 120, 120, 124, 128, 132, 132, 136, 140, 144, 144, 148, 152, 156, 156, 160, 164, 168, 168, 172, 176

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Mar 20 2003

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A080455, A080456, A080457, A080036, A080037.

LinearRecurrence[{1, 0, 0, 1, -1}, {4, 8, 12, 12, 16}, 60] (* Jean-François Alcover, Sep 20 2018 *)

a(n) = 4 + 4*(n-2-(n-4)\4); \\ Michel Marcus, May 06 2016

a(n) = 4 + 4*(n-2-floor((n-4)/4)).
From Chai Wah Wu, Jul 17 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: 4*x*(x^2 + x + 1)/(x^5 - x^4 - x + 1). (End)
From Ilya Gutkovskiy, Jul 17 2016: (Start)
E.g.f.: (3*x + 1)*cosh(x) + (3*x + 2)*sinh(x) - cos(x) - sin(x).
a(n) = (6*n - (-1)^n - 2*sqrt(2)*sin(Pi*n/2+Pi/4) + 3)/2. (End)

A080456 a(1) = a(2) = 2; for n > 2, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.

Original entry on oeis.org

2, 2, 6, 10, 14, 18, 18, 22, 26, 30, 30, 34, 38, 42, 42, 46, 50, 54, 54, 58, 62, 66, 66, 70, 74, 78, 78, 82, 86, 90, 90, 94, 98, 102, 102, 106, 110, 114, 114, 118, 122, 126, 126, 130, 134, 138, 138, 142, 146, 150, 150, 154, 158, 162, 162, 166, 170, 174, 174

Offset: 1

N. J. A. Sloane, Mar 20 2003

nonn

First differences are 4-periodic.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1).

Cf. A080455, A080457, A080458, A080036, A080037.

Join[{2}, LinearRecurrence[{1, 0, 0, 1, -1}, {6, 10, 14, 18, 18}, 60]] (* Jean-François Alcover, Sep 02 2018 *)
CoefficientList[Series[-2*(-1 - 2 x^2 - 2 x^3 - x^4 - 2 x^5 + 2 x^6)/((-1 + x)^2 (1 + x +x^2 + x^3)), {x, 0, 60}],x] (* Stefano Spezia, Sep 02 2018 *)

From Chai Wah Wu, Jul 17 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 6.
G.f.: -2*(-1 - 2*x^2 - 2*x^3 - x^4 - 2*x^5 + 2*x^6)/((-1 + x)^2*(1 + x + x^2 + x^3)). (End)

a(1) = 2 prepended by Stefano Spezia, Sep 04 2018

A080667 a(1)=3; for n>1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.

Original entry on oeis.org

3, 6, 10, 13, 16, 20, 23, 26, 29, 33, 36, 39, 43, 46, 49, 53, 56, 59, 62, 66, 69, 72, 76, 79, 82, 86, 89, 92, 96, 99, 102, 105, 109, 112, 115, 119, 122, 125, 129, 132, 135, 138, 142, 145, 148, 152, 155, 158, 162, 165, 168, 171, 175, 178, 181, 185, 188

Offset: 1

N. J. A. Sloane, Mar 23 2003

nonn

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,3,4,3)-hiccup sequence. - Michael De Vlieger, Jul 29 2025

Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 3.

Cf. A080455, A080456, A080457, A080458, A080036, A080037.

a[1] = 3; a[n_] := a[n] = a[n-1] + If[MemberQ[Array[a, n-1], n], 4, 3];
Array[a, 60] (* Jean-François Alcover, Nov 25 2018 *)

a(n) = floor(n*r + 1/(1+r)) where r = (3 + sqrt(13))/2.

A080460 a(1) = 2; for n > 1, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.

Original entry on oeis.org

2, 2, 6, 10, 14, 14, 18, 22, 26, 26, 30, 34, 38, 38, 42, 46, 50, 50, 54, 58, 62, 62, 66, 70, 74, 74, 78, 82, 86, 86, 90, 94, 98, 98, 102, 106, 110, 110, 114, 118, 122, 122, 126, 130, 134, 134, 138, 142, 146, 146, 150, 154, 158, 158, 162, 166, 170, 170

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Mar 22 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A080455, A080456, A080457, A080458, A080036, A080037.

LinearRecurrence[{1, 0, 0, 1, -1}, {2, 2, 6, 10, 14}, 58] (* Jean-François Alcover, Jan 07 2019 *)

a(n)=4*n - (n-2)\4*4 - 6 \\ Charles R Greathouse IV, Jul 17 2016

a(n) = 2 + 4*(n - 2 - floor((n - 2)/4)).
From Chai Wah Wu, Jul 17 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: 2*x*(x^4 + 2*x^3 + 2*x^2 + 1)/(x^5 - x^4 - x + 1). (End)
From Ilya Gutkovskiy, Jul 17 2016: (Start)
E.g.f.: 2 + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x) + sin(x) + cos(x).
a(n) = (6*n - (-1)^n + 2*sqrt(2)*sin(Pi*n/2 + Pi/4) - 5)/2. (End)

cp's OEIS Frontend

A080578 a(1)=1; for n > 1, a(n) = a(n-1) + 1 if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A080458 a(1)=4; for n>1, a(n)=a(n-1) if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A080456 a(1) = a(2) = 2; for n > 2, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A080667 a(1)=3; for n>1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A080460 a(1) = 2; for n > 1, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula