A369186 The denominators of a series that converges to the Dottie Number (A003957).

1, 3, 12, 260, 5720, 314248, 17255072, 1769058016, 181357735680, 29880655637760, 4923158441956352, 1189676108826729472, 287484053261423565824, 95784714773484796761088, 31913810779214031287095296, 2804341960426298188743438336, 1232120770958699233546743119872

Offset

1,2

Comments

Whittaker's root series formula is applied to 1 - x - x^2/2! + x^4/4! - x^6/6! + ..., which is the Taylor expansion of cos(x) - x. The following infinite series for the Dottie number (D) is obtained: D = 1/1 - 1/3 + 1/12 - 3/260 + 1/5720 + 205/314248 - 4439/17255072 ... . The sequence is formed by the denominators of the series.

Links

Table of n, a(n) for n=1..17.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Formula

a(1)=1;

for n > 1, a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=-1, c(2)=-1/2!, c(3)=0, c(4)=1/4!, c(5)=0, c(6)=-1/6!, and c(n) is the coefficient of x^n in the Taylor expansion of cos(x)-x.

Example

a(1) is the denominator of -1/-1 = 1/1.
a(2) is the denominator of simplified -(-1/2!)/(-1* det ToeplitzMatrix((-1,1),(-1,-1/2!))) = (1/2)/(-3/2) = -1/3.
a(3) is the denominator of the simplified -det ToeplitzMatrix((-1/2!,-1),(-1/2!,0))/(det ToeplitzMatrix((-1,1),(-1,-1/2!))*det ToeplitzMatrix((-1,1,0),(-1,-1/2!,0)) = -(1/4)/((3/2)*-2) = 1/12.

Crossrefs

Cf. A003957.

Sequence in context: A321603 A297564 A280086 * A102945 A300532 A249940

Adjacent sequences: A369183 A369184 A369185 * A369187 A369188 A369189

Keyword

nonn

Author

Raul Prisacariu, Jan 15 2024

Extensions

a(8)-a(17) from Chai Wah Wu, Feb 10 2024

Status

approved