login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099254 Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot). 22
1, 2, 1, -2, -4, -2, 3, 6, 3, -4, -8, -4, 5, 10, 5, -6, -12, -6, 7, 14, 7, -8, -16, -8, 9, 18, 9, -10, -20, -10, 11, 22, 11, -12, -24, -12, 13, 26, 13, -14, -28, -14, 15, 30, 15, -16, -32, -16, 17, 34, 17, -18, -36, -18, 19, 38, 19, -20, -40, -20, 21, 42, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A granny knot sequence.

INVERTi transform of A077855: (1, 3, 6, 11, 20, 36, 64, 133, ...). - Gary W. Adamson, Jan 13 2017

LINKS

A.H.M. Smeets, Table of n, a(n) for n = 0..20000

Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

FORMULA

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

a(n) = 4*sqrt(3)*sin(Pi*n/3 + Pi/3)/9 + 2*(n + 1)*sin(Pi*n/3 + Pi/6)/3.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(n-k+1)*(-1)^k. - Paul Barry, Nov 12 2004

a(n) = 2*cos(2*Pi*(n + 2)/3)*(floor(n/3) + 1)*(-1)^(n+1). - Tani Akinari, Jul 01 2013

a(n) = (1/54)*(18*(n + 2)*(-1)^floor(n/3) + (3*n + 11)*(-1)^floor((n + 1)/3) - 9*(n + 1)*(-1)^floor((n + 2)/3) - 2*(3*n + 8)*(-1)^floor((n + 4)/3)). - John M. Campbell, Dec 23 2016

From A.H.M. Smeets, Sep 13 2018 (start)

a(n) = 2*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) for n >= 4.

a(3*k) = a(3*k+2) = (-1)^k*(k + 1) for k >= 0.

a(3*k+1) = -(-1)^k*2*(k + 1) for k >= 0. (End)

MAPLE

A099254 := proc(n)

option remember ;

if n <= 3 then

op(n+1, [1, 2, 1, -2]) ;

else

2*procname(n-1)-3*procname(n-2)+2*procname(n-3)-procname(n-4) ;

end if;

end proc:

seq(A099254(n), n=0..80) ; # R. J. Mathar, Jul 08 2022

MATHEMATICA

LinearRecurrence[{2, -3, 2, -1}, {1, 2, 1, -2}, 100] (* Jean-François Alcover, Sep 21 2022 *)

PROG

(Python)

a0, a1, a2, a3, n = -2, 1, 2, 1, 3

print(0, a3)

print(1, a2)

print(2, a1)

print(3, a0)

while n < 20000:

a0, a1, a2, a3, n = 2*a0-3*a1+2*a2-a3, a0, a1, a2, n+1

print(n, a0) # A.H.M. Smeets, Sep 13 2018

CROSSREFS

Row sums of array A128502.

Cf. A077855, A076118 (first differences).

Sequence in context: A119538 A068309 A099470 * A186731 A180108 A300417

Adjacent sequences: A099251 A099252 A099253 * A099255 A099256 A099257

KEYWORD

sign,easy

AUTHOR

Paul Barry, Oct 08 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 18:51 EST 2022. Contains 358588 sequences. (Running on oeis4.)