A051951 Second differences of tau(n).

0, -1, 1, -2, 3, -4, 4, -3, 2, -3, 6, -8, 6, -2, 1, -4, 7, -8, 8, -6, 2, -2, 8, -11, 6, -1, 2, -6, 10, -12, 10, -6, 2, 0, 5, -12, 9, -2, 4, -10, 12, -12, 10, -4, -2, 0, 10, -15, 10, -5, 4, -6, 10, -10, 8, -8, 4, -2, 12, -20, 12, 0, -1, -4, 7, -10, 10, -6, 6, -10, 16, -20, 12, 0, -2, -2, 6, -10, 14, -13, 4

Offset: 0

Author

N. J. A. Sloane, Dec 17 1999

Keywords

• sign

Comments

a(0) = 0 assuming tau(0) = 0. - Michael Somos, Mar 05 2014

Examples

			G.f. = - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 4*x^6 - 3*x^7 + 2*x^8 - 3*x^9 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000005, A051950.

Programs

Maple

Taus:= map(numtheory:-tau, [$0..100]):
Taus[3..-1] - 2*Taus[2..-2] + Taus[1..-3]; # Robert Israel, Dec 07 2015

Mathematica

Join[{0},Differences[DivisorSigma[0,Range[90]],2]] (* Harvey P. Dale, Jan 10 2014 *)
a[ n_] := SeriesCoefficient[ (QHypergeometricPFQ[ {x, x}, {x^2, x^2}, x, x^2] - 1) / x, {x, 0, n}]; (* Michael Somos, Mar 05 2014 *)

Formula

G.f.: -1/x + (1 - x)^2*Sum_{k>=1} x^(k-2)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017