Daniele P. Morelli - cp's OEIS frontend

User: Daniele P. Morelli

Daniele P. Morelli has authored 3 sequences.

A296105 a(n) is the number of connected transitive relations over n unlabeled nodes.

1, 2, 5, 25, 157, 1325, 14358, 199763, 3549001, 80673244, 2352747542, 88240542454, 4261209044877, 264988507673267, 21207485269909946, 2182146922863398203

Offset: 0

Daniele P. Morelli, Dec 04 2017

Inverse Euler transform of A091073. Here "connected" means that it is possible to reach any vertex starting from any other vertex by traversing edges in some direction, i.e., not necessarily in the direction in which the edges point, as in weakly connected digraphs.

			a(2) = 5 because there are five connected transitive relations up to isomorphism: a->b with no loops, a->b with a loop on a, a->b with a loop on b, a->b->a with no loops, and a->b->a with loops on both a and b.

Cf. A091073 (all unlabeled transitive relations). For the labeled case, see A245731 (connected labeled transitive relations) and A006905 (all labeled transitive relations).

A091073 = Cases[Import["https://oeis.org/A091073/b091073.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[A091073 // Rest] (* Jean-François Alcover, Dec 29 2019, updated Mar 17 2020 *)

A228551 Number of connected labeled graded posets on n vertices.

Original entry on oeis.org

1, 1, 2, 12, 146, 2820, 79682, 3109932, 163268786, 11373086100, 1049057429762, 128748967088412, 21220651011079346, 4747770003765805380, 1456585782002699624642, 6390825031791150383749864

Offset: 0

Daniele P. Morelli, Aug 25 2013

nonn

Here 'graded' is to be intended as in A001833.

David A. Klarner, The number of graded partially ordered sets, Journal of Combinatorial Theory, vol.6, no.1, pp.12-19, (January-1969).

Cf. A001833.

E.g.f.: 1 + log(F(x)), where F(x) is the e.g.f. of A001833.

A156309 Decimal expansion of the absolute value of the larger solution of (n^2+n)/2 = -1/12. (Real root q of 6n^2 + 6n + 1, the other root being p = -1-q.)

Original entry on oeis.org

2, 1, 1, 3, 2, 4, 8, 6, 5, 4, 0, 5, 1, 8, 7, 1, 1, 7, 7, 4, 5, 4, 2, 5, 6, 0, 9, 7, 4, 9, 0, 2, 1, 2, 7, 2, 1, 7, 6, 1, 9, 9, 1, 2, 4, 3, 6, 4, 9, 3, 6, 5, 6, 1, 9, 9, 0, 6, 9, 8, 8, 3, 6, 7, 5, 8, 0, 1, 1, 1, 6, 3, 8, 4, 8, 5, 3, 3, 3, 2, 7, 1, 5, 3, 1, 4, 2, 3, 0, 2, 2, 0, 7, 1, 2, 5, 2, 3, 7, 3, 8, 7, 3, 9

Offset: 0

Daniele P. Morelli, Feb 07 2009

The formula returning the n-th triangular number (A000217) is (n^2+n)/2. On the other hand, Ramanujan's identity claims that the value of the infinite sum 1+2+3+.... is -1/12. This irrational number is the solution of the equation (n^2+n)/2 = -1/12, that is, the "limit" triangular number.
Equals the Knuth's random generators constant, that is, the ratio c/m in congruence random number generators of the type X_(n+1) = (aX_n +c) mod (m) which minimizes the correlation between successive values. - Stanislav Sykora, Nov 13 2013
It is also the fraction of the full solid angle cut out by a cone having the magic angle (A195696) as its polar angle. - Stanislav Sykora, Nov 13 2013

			The two roots of 6n^2 + 6n + 1 = 0 are -0.21132... and -0.78867513... (Cf. A020769.)

B. Candelpergher, Ramanujan summation of divergent series. Lectures notes in mathematics 2185, Springer 2017.
D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1969, Chapter 3.3.3.

P. J. Cameron and V. Yildiz, Counting false entries in truth tables of bracketed formulas connected by implication, arXiv:1106.4443 [math.CO], 2011.
Michael Penn, What is the radius of 🔴 ?, YouTube video, 2021.

Cf. A000217, A020760, A020769, A165663, A195696, A334843.

First[RealDigits[(3 - Sqrt[3])/6, 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

abs(solve(n=-1/2, 0, 6*n^2+6*n+1)) \\ Michel Marcus, Oct 05 2013

(1 - 1/sqrt(3))/2 = (1 - A020760)/2 = 1/2 - A020769. - R. J. Mathar, Feb 10 2009
Equals - HurwitzZeta(-1, (9 - sqrt(3))/6). - Peter Luschny, Jul 05 2020
Equals (3 - sqrt(3))/6. - Michel Marcus, Jun 10 2021
Equals 1/A165663 = A334843/3. - Hugo Pfoertner, Jun 25 2024

Flipped sign of definition, corrected offset, simplified formula R. J. Mathar, Feb 10 2009

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User: Daniele P. Morelli

A296105 a(n) is the number of connected transitive relations over n unlabeled nodes.

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A228551 Number of connected labeled graded posets on n vertices.

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A156309 Decimal expansion of the absolute value of the larger solution of (n^2+n)/2 = -1/12. (Real root q of 6n^2 + 6n + 1, the other root being p = -1-q.)

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