A257145 a(n) = 5 * floor( (n+2) / 5) - n with a(0) = 1.

1, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0

Offset: 0

Author

Michael Somos, Apr 16 2015

Keywords

• sign
• easy

Comments

Cycle period is 5, {0, -1, -2, 2, 1} after the first five terms. - Robert G. Wilson v, Aug 02 2018

Examples

			G.f. = 1 - x - 2*x^2 + 2*x^3 + x^4 - x^6 - 2*x^7 + 2*x^8 + x^9 - x^11 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).

Crossrefs

Cf. A257143, A253262, A117444.

Programs

Haskell

a257145 0 = 1
a257145 n = div (n + 2) 5 * 5 - n  -- Reinhard Zumkeller, Apr 17 2015

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1-x^2)^2/(1-x^5))); // G. C. Greubel, Aug 02 2018

Mathematica

a[ n_] := If[ n==0, 1, -Mod[ n, 5, -2]];
a[ n_] := If[ n==0, 1, Sign[n] SeriesCoefficient[ (1 - x) * (1 - x^2)^2 / (1 - x^5), {x, 0, Abs@n}]];
CoefficientList[Series[(1-x)*(1-x^2)^2/(1-x^5), {x,0,60}], x] (* G. C. Greubel, Aug 02 2018 *)
a[n_] := 5 Floor[(n + 2)/5] - n; Array[a, 77, 0] (* or *)
CoefficientList[ Series[(x - 1)^2 (x + 1)^2/(x^4 + x^3 + x^2 + x + 1), {x, 0, 76}], x] (* or *)
LinearRecurrence[{-1, -1, -1, -1}, {1, -1, -2, 2, 1, 0}, 76] (* Robert G. Wilson v, Aug 02 2018*)

PARI

{a(n) = if( n==0, 1, (n+2) \ 5 * 5 - n)};

PARI

{a(n) = if( n==0, 1, [0, -1, -2, 2, 1][n%5 + 1])};

PARI

{a(n) = if( n==0, 1, sign(n) * polcoeff( (1 - x) * (1 - x^2)^2 / (1 - x^5) + x * O(x^abs(n)), abs(n)))};

PARI

x='x+O('x^60); Vec((1-x)*(1-x^2)^2/(1-x^5)) \\ G. C. Greubel, Aug 02 2018

Formula

Euler transform of length 5 sequence [-1, -2, 0, 0, 1].

a(5*n) = 0 for all n in Z except n=0.

a(n) = -a(-n) for all n in Z except n=0.

a(n) = a(n+5) for all n in Z except n=-5 or n=0.

Convolution inverse is A257143.

G.f.: (1 - x) * (1 - x^2)^2 / (1 - x^5).

G.f.: (1 - 2*x^2 + x^4) / (1 + x + x^2 + x^3 + x^4).

a(n) = -A117444(n), n>0. - R. J. Mathar, Oct 05 2017