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A009531 Expansion of the e.g.f. sin(x)*(1+x). 12
0, 1, 2, -1, -4, 1, 6, -1, -8, 1, 10, -1, -12, 1, 14, -1, -16, 1, 18, -1, -20, 1, 22, -1, -24, 1, 26, -1, -28, 1, 30, -1, -32, 1, 34, -1, -36, 1, 38, -1, -40, 1, 42, -1, -44, 1, 46, -1, -48, 1, 50, -1, -52, 1, 54, -1, -56, 1, 58, -1, -60, 1, 62, -1, -64, 1, 66, -1, -68, 1, 70, -1, -72, 1, 74, -1, -76, 1, 78, -1, -80 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16384

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

FORMULA

There's an obvious formula for the n-th term!

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

abs(a(n)) = Sum_{k=0..floor((n-1)/2)} (C(n-k-1, k) mod 2)*(-1)^k*2^A000120(n-2k-1). - Paul Barry, Jan 06 2005

a(n) = (n^(n+1) mod (n+1)) * (-1)^[(n-1)/2] = a(n-1)-a(n-2)+(-1)^n*a(n-1) = -2a(n-2)-a(n-4). - Henry Bottomley, May 07 2005

a(n+2) is the Hankel transform of A086622(n+1). - Paul Barry, Nov 06 2007

E.g.f.: sin(x)*(1+x)=x*Q(0); Q(k)=1+x/(1-x/(x-2*(k+1)*(2k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011

a(n) = sin(Pi*n/2)-n*cos(Pi*n/2). - Vaclav Kotesovec, Oct 03 2014

a(n) = (((2*n+3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4)+((2*n-1-(-1)^n)/2)*(-1)^((6*n+5-(-1)^n)/4))/2. - Luce ETIENNE, Jul 18 2015

MATHEMATICA

CoefficientList[Series[x*(1+x)^2/(1+x^2)^2, {x, 0, 100}], x] (* Vaclav Kotesovec, Oct 03 2014 *)

PROG

(PARI) concat(0, Vec(x*(1+x)^2/(1+x^2)^2 + O(x^80))) \\ Michel Marcus, Oct 03 2014

(PARI) A009531(n) = (((n^(n+1)) % (n+1)) * ((-1)^((n-1)\2))); \\ Antti Karttunen, Nov 02 2017, after Henry Bottomley's formula.

(PARI) A009531(n) = (lift(Mod(n, n+1)^(n+1)) * ((-1)^((n-1)\2))); \\ (like above, but quicker) - Antti Karttunen, Nov 02 2017

(MAGMA) [(((2*n+3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n) div 4)+((2*n-1-(-1)^n) div 2)*(-1)^((6*n+5-(-1)^n) div 4))/2: n in [0..90]]; // Vincenzo Librandi, Jul 19 2015

CROSSREFS

Cf. A009001, A029578, A049240.

Sequence in context: A083258 A083259 A214060 * A124625 A318775 A317500

Adjacent sequences:  A009528 A009529 A009530 * A009532 A009533 A009534

KEYWORD

sign,easy

AUTHOR

R. H. Hardin

STATUS

approved

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Last modified May 25 19:13 EDT 2019. Contains 323576 sequences. (Running on oeis4.)