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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.) 2

%I #37 Jun 11 2021 12:17:27

%S 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,

%T 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,

%U 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

%N 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.)

%C 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.

%C 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

%C 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

%D B. Candelpergher, Ramanujan summation of divergent series. Lectures notes in mathematics 2185, Springer 2017.

%D D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1969, Chapter 3.3.3.

%H P. J. Cameron and V. Yildiz, <a href="https://arxiv.org/abs/1106.4443">Counting false entries in truth tables of bracketed formulas connected by implication</a>, arXiv:1106.4443 [math.CO], 2011.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=RKTqKe4M5wM">What is the radius of 🔴 ?</a>, YouTube video, 2021.

%F (1 - 1/sqrt(3))/2 = (1 - A020760)/2 = 1/2 - A020769. - _R. J. Mathar_, Feb 10 2009

%F Equals - HurwitzZeta(-1, (9 - sqrt(3))/6). - _Peter Luschny_, Jul 05 2020

%F Equals (3 - sqrt(3))/6. - _Michel Marcus_, Jun 10 2021

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

%o (PARI) abs(solve(n=-1/2, 0, 6*n^2+6*n+1)) \\ _Michel Marcus_, Oct 05 2013

%Y Cf. A000217, A020769, A195696.

%K nonn,cons

%O 0,1

%A _Daniele P. Morelli_, Feb 07 2009

%E Flipped sign of definition, corrected offset, simplified formula _R. J. Mathar_, Feb 10 2009

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)