A368862 Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).

-1, -3, 1, 1, -1, 5, -19, -9, 41, -103, 17, 289, -169, 331, -689, -4991, 3999, 7833, -6509, 21827, -22165, -87637, 119441, -190981, -152513, 1482023, -425985, -1045091, 1071237, -14108791, 5845271, 39852203, -35832801, 54451699, 44061359, -435442725, 261309855, -22217917

Offset: 1

Author

Raul Prisacariu, Jan 08 2024

Keywords

• sign

Comments

Whittaker's root series formula is applied to 1 + Sum_{k>=1} prime(k) x^k. The following infinite series that converges to the negative inverse of Backhouse's constant (-x) is obtained:

x = -1/(1*2) - 3/(2*1) + 1/(1*1) + 1/(1*2) - 1/(2*3) + 5/(3*7) - 19/(7*10) - 9/(10*13) + 41/(13*21) - 103/(21*26) + 17/(26*33) + 289/(33*53) ...

The denominators of the infinite series are obtained by multiplying the absolute values of 2 consecutive terms from the sequence A030018.

Examples

			a(1) = -1;
a(2) = -3;
a(3) = -det ToeplitzMatrix((3,2),(3,5)) = 1;
a(4) = -det ToeplitzMatrix((3,2,1),(3,5,7)) = 1;
a(5) = -det ToeplitzMatrix((3,2,1,0),(3,5,7,11)) = -1;
a(6) = -det ToeplitzMatrix((3,2,1,0,0),(3,5,7,11,13)) = 5;
a(7) = -det ToeplitzMatrix((3,2,1,0,0,0),(3,5,7,11,13,17)) = -19.
		

Links

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

Crossrefs

Cf. A088751, A030018, A072508.

Formula

a(1) = -1.

For n > 1, a(n) = -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n))), where c(0)=1 and c(n) is the n-th prime number.

Extensions

a(21)-a(38) from Stefano Spezia, Jan 09 2024