A081834 -id:A081834 - cp's OEIS frontend

A080590 a(1)=1; for n>1, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

1, 5, 9, 13, 16, 20, 24, 28, 31, 35, 39, 43, 46, 50, 54, 57, 61, 65, 69, 72, 76, 80, 84, 87, 91, 95, 99, 102, 106, 110, 113, 117, 121, 125, 128, 132, 136, 140, 143, 147, 151, 155, 158, 162, 166, 169, 173, 177, 181, 184, 188, 192, 196, 199, 203, 207

Offset: 1

N. J. A. Sloane, Mar 23 2003

nonn

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,3,4)-hiccup sequence. - Michael De Vlieger, Jul 29 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 9.
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. A003151, A080455-A080458, A080036, A080037.

Cf. A081834.

a080590 n = a080590_list !! (n-1)
a080590_list = 1 : f 2 [1] where
   f x zs@(z:_) = y : f (x + 1) (y : zs) where
     y = z + (if x `elem` zs then 3 else 4)
-- Reinhard Zumkeller, Sep 26 2014

a[n_] := a[n] = If[n == 1, 1, If[MemberQ[Array[a, n-1], n], a[n-1]+3, a[n-1]+4]]; Array[a, 56] (* Jean-François Alcover, Oct 05 2018 *)

A081840 a(1)=0, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

Original entry on oeis.org

0, 4, 8, 11, 15, 19, 23, 26, 30, 34, 37, 41, 45, 49, 52, 56, 60, 64, 67, 71, 75, 79, 82, 86, 90, 93, 97, 101, 105, 108, 112, 116, 120, 123, 127, 131, 134, 138, 142, 146, 149, 153, 157, 161, 164, 168, 172, 176, 179, 183, 187, 190, 194, 198, 202, 205, 209, 213, 217

Offset: 1

Benoit Cloitre, Apr 11 2003

nonn

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 9.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 3.

Cf. A080353, A080580, A080590, A080600, A080652, A080754, A081834.

Block[{c, k}, c[] := False; k = 0; c[0] = True; {k}~Join~Reap[Do[If[c[n], k += 3, k += 4]; c[k] = True; Sow[k], {n, 2, 120}]][[-1, 1]] ] (* _Michael De Vlieger, Jul 02 2025 *)

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

A081841 a(1)=0, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

Original entry on oeis.org

0, 2, 4, 7, 9, 11, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 40, 43, 45, 48, 50, 52, 55, 57, 60, 62, 65, 67, 69, 72, 74, 77, 79, 81, 84, 86, 89, 91, 94, 96, 98, 101, 103, 106, 108, 110, 113, 115, 118, 120, 123, 125, 127, 130, 132, 135, 137, 139, 142, 144, 147, 149, 151

Offset: 1

Benoit Cloitre, Apr 11 2003

nonn

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,0,3,2)-hiccup sequence. - Michael De Vlieger, Jul 29 2025

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 7.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 3, 8.

Cf. A080353, A080580, A080590, A080600, A080652, A080754, A081834, A081840

Block[{c, k}, c[] := False; k = 0; c[0] = True; {k}~Join~Reap[Do[If[c[n], k += 3, k += 2]; c[k] = True; Sow[k], {n, 2, 120}]][[-1, 1]] ] (* _Michael De Vlieger, Jul 02 2025 *)

a(1)=0; for n>=1 a(n)=floor(r*n-(4*r-1)/(r+1)) where r=1+sqrt(2)

A081842 a(1)=0, 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

0, 3, 7, 10, 13, 16, 20, 23, 26, 30, 33, 36, 40, 43, 46, 50, 53, 56, 59, 63, 66, 69, 73, 76, 79, 83, 86, 89, 92, 96, 99, 102, 106, 109, 112, 116, 119, 122, 125, 129, 132, 135, 139, 142, 145, 149, 152, 155, 158, 162, 165, 168, 172, 175, 178, 182, 185, 188, 192, 195

Offset: 1

Benoit Cloitre, Apr 11 2003

nonn

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 7.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 3.

Cf. A080353, A080580, A080590, A080600, A080652, A080754, A081834, A081840, A081841.

Module[{seq={0},n=2},Do[If[MemberQ[seq,n],AppendTo[seq,Last[seq]+4],AppendTo[seq, Last[seq]+3]];n++,{60}];seq] (* Harvey P. Dale, Nov 20 2012 *)

a(n) = floor(r*n-(4*r-1)/(r+1)) where r = (1/2)*(3+sqrt(13)).

A081839 a(1)=0, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

Original entry on oeis.org

0, 5, 10, 15, 19, 24, 29, 34, 39, 43, 48, 53, 58, 63, 67, 72, 77, 82, 86, 91, 96, 101, 106, 110, 115, 120, 125, 130, 134, 139, 144, 149, 154, 158, 163, 168, 173, 178, 182, 187, 192, 197, 201, 206, 211, 216, 221, 225, 230, 235, 240, 245, 249, 254, 259, 264, 269

Offset: 1

Benoit Cloitre, Apr 11 2003

nonn

In the Fokkink-Joshi paper, this sequence is the Cloitre (0,0,4,5)-hiccup sequence. - Michael De Vlieger, Jul 29 2025

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 3.

Cf. A081834, A081835, A081840, A081841, A081842, A081843.

cloitreH[j_, x_, y_, z_, w_ : 120] := Module[{c, k}, c[] := False; k = x; c[x] = True; {x}~Join~Reap[Do[If[c[n - j], k += y, k += z]; c[k] = True; Sow[k], {n, 2, w}] ][[-1, 1]] ]; cloitreH[0, 0, 4, 5] (* _Michael De Vlieger, Jul 29 2025 *)

a(n) = floor(r*n-(3*r+1)/(r-1)) where r = (1/2) *(5+sqrt(21)) = 4.79128784747792...

cp's OEIS Frontend

A080590 a(1)=1; for n>1, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

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A081840 a(1)=0, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

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A081841 a(1)=0, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

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A081842 a(1)=0, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.

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A081839 a(1)=0, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

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