A140440 -id:A140440 - cp's OEIS frontend

A190436 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427 - A190430
(golden ratio,3,0):  A140397 - A190400
(golden ratio,3,1):  A140431 - A190435
(golden ratio,3,2):  A140436 - A190439
(golden ratio,4,c):  A140440 - A190461

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190437, A190438, A190439, A190440.

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A302253 Positions of 3 in A190436.

Original entry on oeis.org

8, 21, 29, 42, 55, 63, 76, 97, 110, 118, 131, 144, 152, 165, 186, 199, 207, 220, 241, 254, 262, 275, 288, 296, 309, 330, 343, 351, 364, 377, 385, 398, 406, 419, 432, 440, 453, 474, 487, 495, 508, 521, 529, 542, 563, 576, 584, 597, 618, 631, 639, 652, 665, 673, 686, 707, 720, 728

Offset: 1

G. C. Greubel, Apr 04 2018

nonn

Write a(n) = [(bn+c)r] - b[nr] - [cr]. If r>0 and b and c are integers satisfying b >= 2 and 0 <= c <= b-1, then 0 <= a(n) <= b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A140440-A190461

G. C. Greubel, Table of n, a(n) for n = 1..20000

Cf. A190436, A190437, A190438, A190439.

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 500}] (* A190436 *)
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A126757 Number of n-node connected graphs with no cycles of length less than 5.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 18, 47, 137, 464, 1793, 8167, 43645, 275480, 2045279, 17772647, 179593823, 2098423758, 28215583324, 434936005284, 7662164738118

Offset: 1

N. J. A. Sloane, Feb 18 2007

Keith M. Briggs, Combinatorial Graph Theory
CombOS - Combinatorial Object Server, Generate graphs

Cf. A006787, A140440.

A006787 = Cases[Import["https://oeis.org/A006787/b006787.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A006787] (* Jean-François Alcover, May 13 2019, updated Mar 17 2020 *)

This is the inverse Euler transform of A006787. - Conjectured by Vladeta Jovovic, Jun 16 2008, proved by Max Alekseyev and Brendan McKay, Jun 17 2008

Definition corrected by Max Alekseyev and Brendan McKay, Jun 17 2008

a(20)-a(21) using Brendan McKay's extension to A006787 by Alois P. Heinz, Mar 11 2018

A366224 Number of unlabeled 3-connected graphs on n vertices with girth at least 5.

Original entry on oeis.org

1, 0, 2, 4, 23, 149, 1670, 23882, 422194, 8544496, 195291551

Offset: 10

Brendan McKay, Oct 05 2023

			The smallest such graph is the Petersen graph on 10 vertices.

Cf. A006290, A140440 (2-connected graphs with girth at least 5),

A366225 (3-connected graphs with girth at least 6).

A345324 Number of unlabeled 2-connected simple graphs with girth at least 6.

Original entry on oeis.org

1, 1, 2, 3, 8, 17, 55, 166, 675, 2977, 15553, 90159, 586769, 4197170, 32919812, 281235872, 2609796620

Offset: 6

Brendan McKay, Jun 14 2021

The girth is the length of the shortest cycle, so girth at least 6 means there are no cycles of length 3, 4 or 5.

A345246 for girth at least 4.

A140440 for girth at least 5.

cp's OEIS Frontend

A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

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A302253 Positions of 3 in A190436.

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Mathematica

A126757 Number of n-node connected graphs with no cycles of length less than 5.

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Mathematica

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A366224 Number of unlabeled 3-connected graphs on n vertices with girth at least 5.

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Examples

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A345324 Number of unlabeled 2-connected simple graphs with girth at least 6.

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Author

Keywords

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A190436 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.