A190405 - OEIS

%I #41 Dec 08 2020 02:36:07

%S 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 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 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 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 See A190404.

%C Binary expansion is .1010010001... (A023531). - _Rick L. Shepherd_, Jan 05 2014

%C From _Amiram Eldar_, Dec 07 2020: (Start)

%C This constant is not a quadratic irrational (Duverney, 1995).

%C The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)

%H Danny Rorabaugh, <a href="/A190405/b190405.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 0.64163256065515386629...

%t RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* _Jean-François Alcover_, Feb 12 2013 *)

%t RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* _Harvey P. Dale_, Oct 18 2013 *)

%t (* See also A190404 *)

%o (Sage)

%o def A190405(b):  # Generate the constant with b bits of precision

%o     return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)

%o A190405(409) # _Danny Rorabaugh_, Mar 25 2015

%o (PARI) th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)

%o th2(sqrt(.5))/2^(7/8)-1 \\ _Charles R Greathouse IV_, Jun 06 2016

%Y A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).

%Y A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).

%Y A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).

%Y A190407: Sum_{k>=1} (1/2)^V(k), where V = A058331 (1 + 2*k^2).

%Y Cf. A000079.

%K nonn,cons,easy

%O 0,1

%A _Clark Kimberling_, May 10 2011