A274922 a(n) = (-1)^n * n if n>0, a(0) = 1.

1, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59

Offset

0,3

Comments

This is a divisibility sequence.

Links

Table of n, a(n) for n=0..59.

Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Formula

Euler transform of length 3 sequence [-1, 2, -1].

a(n) = -b(n) where b() is multiplicative with b(2^e) = -(2^e) if e>0, b(p^e) = p^e otherwise.

E.g.f.: 1 - x * exp(-x).

G.f.: (1 + x + x^2) / (1 + 2*x + x^2).

G.f.: (1 - x) * (1 - x^3) / (1 - x^2)^2.

G.f.: 1 / (1 + x / (1 + x / (1 - x / (1 + x)))).

G.f.: 1 - x / (1 + x)^2 = 1 - x /(1 - x)^2 + 4*x^2 / (1 - x^2)^2.

a(n) = (-1)^n * A028310(n). a(2*n) = A004277(n). a(2*n + 1) = - A005408(n).

Convolution inverse of A106510.

A060576(n) = Sum_{k=0..n} binomial(n, k) * a(k).

A028310(n) = Sum_{k=0..n} binomial(n+1, k+1) * a(k).

a(n) = A038608(n), n>0. - R. J. Mathar, May 25 2020

Example

G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 5*x^5 + 6*x^6 - 7*x^7 + 8*x^8 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n n];
a[ n_] := SeriesCoefficient[ (1 + x + x^2) / (1 + 2*x + x^2), {x, 0, n}];
LinearRecurrence[{-2, -1}, {1, -1, 2}, 60] (* Harvey P. Dale, Mar 30 2019 *)

Prog

(PARI) {a(n) = if( n<1, n==0, (-1)^n * n)};
(PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x + x^2) / (1 + 2*x + x^2) + x * O(x^n), n))};
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x+x^2)/(1+2*x+x^2))); // G. C. Greubel, Jul 29 2018

Crossrefs

Cf. A004277, A005408, A028310, A060576, A106510.

Sequence in context: A167976 A024000 A181983 * A097141 A160356 A001478

Adjacent sequences: A274919 A274920 A274921 * A274923 A274924 A274925

Keyword

sign,easy

Author

Michael Somos, Dec 28 2016

Status

approved