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 A190405 #64 Jun 11 2025 21:48:14
%S A190405 6,4,1,6,3,2,5,6,0,6,5,5,1,5,3,8,6,6,2,9,3,8,4,2,7,7,0,2,2,5,4,2,9,4,
%T A190405 3,4,2,2,6,0,6,1,5,3,7,9,5,6,7,3,9,7,4,7,8,0,4,6,5,1,6,2,2,3,8,0,1,4,
%U A190405 4,6,0,3,7,3,3,3,5,1,7,7,5,6,0,0,3,6,4,1,7,1,6,2,3,3,5,9,1,3,3,0,8,6,0,8,9,7,3,5,3,1,6,3,4,3,6,1,9,4,6,1
%N A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.
%C A190405 See A190404.
%C A190405 Binary expansion is .1010010001... (A023531). - _Rick L. Shepherd_, Jan 05 2014
%C A190405 From _Amiram Eldar_, Dec 07 2020: (Start)
%C A190405 This constant is not a quadratic irrational (Duverney, 1995).
%C A190405 The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)
%H A190405 Danny Rorabaugh, <a href="/A190405/b190405.txt">Table of n, a(n) for n = 0..10000</a>
%H A190405 Daniel Duverney, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k62053342/f33.item">Sommes de deux carrés et irrationalité de valeurs de fonctions thêta</a>, Comptes rendus de l'Académie des sciences, Série 1, Mathématique, Vol. 320, No. 9 (1995), pp. 1041-1044.
%e A190405 0.64163256065515386629...
%t A190405 RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* _Jean-François Alcover_, Feb 12 2013 *)
%t A190405 RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* _Harvey P. Dale_, Oct 18 2013 *)
%t A190405 (* See also A190404 *)
%o A190405 (Sage)
%o A190405 def A190405(b): # Generate the constant with b bits of precision
%o A190405 return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)
%o A190405 A190405(409) # _Danny Rorabaugh_, Mar 25 2015
%o A190405 (PARI) th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
%o A190405 th2(sqrt(.5))/2^(7/8)-1 \\ _Charles R Greathouse IV_, Jun 06 2016
%Y A190405 A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).
%Y A190405 A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).
%Y A190405 A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).
%Y A190405 A190407: Sum_{k>=1} (1/2)^V(k), where V = A058331 (1 + 2*k^2).
%Y A190405 Cf. A000079.
%K A190405 nonn,cons,easy
%O A190405 0,1
%A A190405 _Clark Kimberling_, May 10 2011