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

Author

Raul Prisacariu, Jan 15 2024

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A003957.

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.

Extensions

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