cp's OEIS Frontend

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.)

%I A156309 #44 Jun 27 2024 10:10:08
%S A156309 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 A156309 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 A156309 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 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.)
%C A156309 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 A156309 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 A156309 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 A156309 B. Candelpergher, Ramanujan summation of divergent series. Lectures notes in mathematics 2185, Springer 2017.
%D A156309 D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1969, Chapter 3.3.3.
%H A156309 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 A156309 Michael Penn, <a href="https://www.youtube.com/watch?v=RKTqKe4M5wM">What is the radius of 🔴 ?</a>, YouTube video, 2021.
%F A156309 (1 - 1/sqrt(3))/2 = (1 - A020760)/2 = 1/2 - A020769. - _R. J. Mathar_, Feb 10 2009
%F A156309 Equals - HurwitzZeta(-1, (9 - sqrt(3))/6). - _Peter Luschny_, Jul 05 2020
%F A156309 Equals (3 - sqrt(3))/6. - _Michel Marcus_, Jun 10 2021
%F A156309 Equals 1/A165663 = A334843/3. - _Hugo Pfoertner_, Jun 25 2024
%e A156309 The two roots of 6n^2 + 6n + 1 = 0 are -0.21132... and -0.78867513... (Cf. A020769.)
%t A156309 First[RealDigits[(3 - Sqrt[3])/6, 10, 100]] (* _Paolo Xausa_, Jun 25 2024 *)
%o A156309 (PARI) abs(solve(n=-1/2, 0, 6*n^2+6*n+1)) \\ _Michel Marcus_, Oct 05 2013
%Y A156309 Cf. A000217, A020760, A020769, A165663, A195696, A334843.
%K A156309 nonn,cons
%O A156309 0,1
%A A156309 _Daniele P. Morelli_, Feb 07 2009
%E A156309 Flipped sign of definition, corrected offset, simplified formula _R. J. Mathar_, Feb 10 2009