

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 = 1q.)


2



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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

The formula returning the nth 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


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, AddisonWesley, 1969, Chapter 3.3.3.


LINKS



FORMULA

Equals  HurwitzZeta(1, (9  sqrt(3))/6).  Peter Luschny, Jul 05 2020


EXAMPLE

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


MATHEMATICA

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


PROG

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


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS

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


STATUS

approved



