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

0,1

COMMENTS

The formula returning the n-th 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, Addison-Wesley, 1969, Chapter 3.3.3.

LINKS

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

FORMULA

(1-1/sqrt(3))/2 = (1-A020760)/2 = 1/2-A020769. - 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

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Last modified May 23 07:10 EDT 2017. Contains 286909 sequences.