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 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

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