

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).


1



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
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OFFSET

0,1


COMMENTS

The formula returning the nth triangular number (that is, 1+2+3+...+n) 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

D. E. Knuth, The Art of Computer Programming, Vol. 2, AddisonWesley, 1969, Chapter 3.3.3.


LINKS

Table of n, a(n) for n=0..103.


FORMULA

(11/sqrt(3))/2 = (1A020760)/2 = 1/2A020769.  R. J. Mathar, Feb 10 2009


EXAMPLE

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


PROG

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


CROSSREFS

Cf. A000217.
Sequence in context: A071872 A141038 A240055 * A205115 A093623 A156025
Adjacent sequences: A156306 A156307 A156308 * A156310 A156311 A156312


KEYWORD

nonn,cons


AUTHOR

Daniele P. Morelli, Feb 07 2009


EXTENSIONS

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


STATUS

approved



