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.)

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

Author

Daniele P. Morelli, Feb 07 2009

Keywords

• nonn
• cons

Comments

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

Examples

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

References

• 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.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

(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

Extensions

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