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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
 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 (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, Addison-Wesley, 1969, Chapter 3.3.3. LINKS Table of n, a(n) for n=0..103. P. J. Cameron and V. Yildiz, Counting false entries in truth tables of bracketed formulas connected by implication, arXiv:1106.4443 [math.CO], 2011. Michael Penn, What is the radius of ðŸ”´ ?, YouTube video, 2021. FORMULA (1 - 1/sqrt(3))/2 = (1 - A020760)/2 = 1/2 - A020769. - R. J. Mathar, Feb 10 2009 Equals - HurwitzZeta(-1, (9 - sqrt(3))/6). - Peter Luschny, Jul 05 2020 Equals (3 - sqrt(3))/6. - Michel Marcus, Jun 10 2021 Equals 1/A165663 = A334843/3. - Hugo Pfoertner, Jun 25 2024 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 Cf. A000217, A020760, A020769, A165663, A195696, A334843. Sequence in context: A071872 A141038 A240055 * A205115 A334369 A093623 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 August 8 03:17 EDT 2024. Contains 375018 sequences. (Running on oeis4.)