This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143820 #54 Nov 01 2024 23:47:28
%S A143820 1,0,4,1,8,6,5,3,5,5,0,9,8,9,0,9,8,4,6,3,0,1,3,3,6,6,1,5,0,2,1,5,2,7,
%T A143820 3,8,7,6,9,7,0,8,3,5,7,1,7,2,4,1,6,3,4,5,9,5,4,5,7,3,9,2,5,5,4,2,3,5,
%U A143820 5,1,7,4,1,1,6,1,0,7,4,0,2,9,5,9,2,8,6,2,6,7,3,9,3,0,1,0,0,6,5,5,2
%N A143820 Decimal expansion of the constant 1/1! + 1/4! + 1/7! + ...
%C A143820 Define a sequence R(n) of real numbers by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(2) - R(1); the decimal expansions of R(0) = 1 + 1/3! + 1/6! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1).
%C A143820 R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
%C A143820 ========================================
%C A143820 | linear combination of
%C A143820 R(n) | R(0) R(1) R(2) - R(1)
%C A143820 ========================================
%C A143820 R(3) | 1 1 3
%C A143820 R(4) | 6 2 7
%C A143820 R(5) | 25 11 16
%C A143820 R(6) | 91 66 46
%C A143820 R(7) | 322 352 203
%C A143820 R(8) | 1232 1730 1178
%C A143820 R(9) | 5672 8233 7242
%C A143820 R(10) | 32202 39987 43786
%C A143820 ...
%C A143820 The column entries are from A143815, A143816 and A143817.
%C A143820 The Abraham Ungar 1982 article defines H_{n,r}(z) = Sum_{k>=0} z^(nk+r)/(nk+r)! as equation (1). The constant is H_{3,1}(1). In equation (13) H_{3,1}(x) = (exp(x) + 2 * exp(-x/2) * cos(sqrt(3)/2*x - 2*Pi/3))/3. In equation (12) the expression H_{3,1}(x) = (e^x + q_2 e^{q_1 x} + q_1 e^{q_2 x})/3 where q_1 = (-1 + I sqrt(3))/2 and q_2 = (-1 - I sqrt(3))/2 is given for H_{3,2}(x) instead. - _Michael Somos_, Nov 01 2024
%H A143820 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2011). See p. 76.
%H A143820 Abraham Ungar, <a href="https://doi.org/10.1080/00029890.1982.11995514">Generalized Hyperbolic Functions</a>, The American Mathematical Monthly, Volume 89, 1982, 688-691.
%F A143820 Equals (exp(1) + w^2*exp(w) + w*exp(w^2))/3, where w = exp(2*Pi*i/3).
%F A143820 A143819 + A143820 + A143821 = exp(1).
%F A143820 Equals Sum_{n>=0} 1/(3*n+1)!. - _Michal Paulovic_, Aug 20 2023
%F A143820 Continued fraction: 1 + 1/(24 - 24/(211 - 210/(721 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n - 1)*(3*n)*(3*n + 1) for n >= 1. Cf. A346441. - _Peter Bala_, Feb 22 2024
%F A143820 Equals (exp(1) + 2*exp(-1/2)*cos(sqrt(3)/2-2*Pi/3))/3. [Ungar, p.690] - _Michael Somos_, Nov 01 2024
%e A143820 1.041865355098909...
%p A143820 Digits:=101: evalf(sum(1/(3*n+1)!, n=0..infinity)); # _Michal Paulovic_, Aug 20 2023
%t A143820 RealDigits[ N[ (-Cos[Sqrt[3]/2] + E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E]), 105]][[1]] (* _Jean-François Alcover_, Nov 08 2012 *)
%o A143820 (PARI) suminf(n=0,1/(3*n+1)!) \\ _Michel Marcus_, Aug 20 2023
%Y A143820 Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143821, A346441.
%K A143820 cons,easy,nonn
%O A143820 1,3
%A A143820 _Peter Bala_, Sep 03 2008
%E A143820 Offset corrected by _R. J. Mathar_, Feb 05 2009