A326582 A signed variant of A309132.

1, 1, 1, 4, 1, 9, 1, 16, 27, 25, 1, 36, 1, 49, -75, 64, 1, 81, -1, 100, 49, 121, -1, 144, 125, 169, -243, 196, 1, 225, -1, 256, 363, 289, -1225, 324, 1, 361, -169, 400, 1, 441, -1, 484, 135, 529, -1, 576, 343, 625, -867, 676, 1, 729, -3025, 784, 361, 841, -1

Offset: 0

Author

Peter Luschny, Jul 15 2019

Keywords

• sign

Comments

See A309132 for background and conjectures.

Crossrefs

Cf. A309132, A157779, A157780.

Programs

Maple

nB := n -> numer(bernoulli(n-1,1/2)): dB := n -> denom(bernoulli(n-1,1/2)):
R := n -> n/(nB(n) + dB(n)/n): a := n -> numer(R(n+1)/4^irem(n,2)):
seq(a(n), n=0..58);

Formula

a(n) = numerator(R(n+1)/4^mod(n,2)) with R(n) = n/(nB(n) + dB(n)/n) and nB(n) = numerator(B(n-1, 1/2)), dB(n) = denominator(B(n-1, 1/2)) where B(n, x) denotes the Bernoulli polynomials.

|a(2*n)| = A309132(2*n + 1) for n >= 0.

a(2*n+1) = (n + 1)^2 for n >= 0.