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 A319373 a(n) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 - ... + (up to n). 9
 1, 2, -1, -10, -5, 20, 13, -36, -27, 54, 43, -78, -65, 104, 89, -136, -119, 170, 151, -210, -189, 252, 229, -300, -275, 350, 323, -406, -377, 464, 433, -528, -495, 594, 559, -666, -629, 740, 701, -820, -779, 902, 859, -990, -945, 1080, 1033, -1176, -1127 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In general, for alternating sequences that multiply the first k natural numbers, and subtract/add the products of the next k natural numbers (preserving the order of operations up to n), we have a(n) = (-1)^floor(n/k) * Sum_{i=1..k-1} (1-sign((n-i) mod k)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/k)+1) * (1-sign(i mod k)) * (Product_{j=1..k} (i-j+1)). Here k=2. An alternating version of A228958. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (1,-3,3,-3,3,-1,1). FORMULA a(n) = (cos(n*Pi/2)*(1-n-n^2) + sin(n*Pi/2)*(1+3*n-n^2) - 1)/2. From Colin Barker, Sep 18 2018: (Start) G.f.: x*(1 + x - 6*x^3 - x^4 + x^5) / ((1 - x)*(1 + x^2)^3). a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6) + a(n-7) for n>7. (End) a(n) = (-1 + (-1)^((n-1)*n/2))/2 + (-2 + (-1)^n)*(-1)^(n*(n+1)/2)*n/2 - (-1)^((n-1)*n/2)*n^2/2. - Bruno Berselli, Sep 25 2018 EXAMPLE a(1) = 1; a(2) = 1*2 = 2; a(3) = 1*2 - 3 = -1; a(4) = 1*2 - 3*4 = -10; a(5) = 1*2 - 3*4 + 5 = -5; a(6) = 1*2 - 3*4 + 5*6 = 20; a(7) = 1*2 - 3*4 + 5*6 - 7 = 13; a(8) = 1*2 - 3*4 + 5*6 - 7*8 = -36; a(9) = 1*2 - 3*4 + 5*6 - 7*8 + 9 = -27; a(10) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 = 54; a(11) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11 = 43; a(12) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 = -78; a(13) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13 = -65; a(14) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 = 104; a(15) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 - 15 = 89; etc. MATHEMATICA Table[(Cos[n Pi/2] (1 - n - n^2) + Sin[n Pi/2] (1 + 3 n - n^2) - 1)/2, {n, 50}] a[n_] := (-1)^Floor[n/2] Sum[(1 - Sign[Mod[n - i, 2]]) Product[n - j + 1, {j, 1, i}], {i, 1, 1}] + Sum[(-1)^(Floor[i/2] + 1) (1 - Sign[Mod[i, 2]]) Product[i - j + 1, {j, 1, 2}], {i, 1, n}]; Array[a, 30] (* Stefano Spezia, Sep 23 2018 *) PROG (PARI) Vec(x*(1 + x - 6*x^3 - x^4 + x^5) / ((1 - x)*(1 + x^2)^3) + O(x^50)) \\ Colin Barker, Sep 18 2018 CROSSREFS Cf. A093361, A228958, A305189. For similar sequences, see: A001057 (k=1), this sequence (k=2), A319543 (k=3), A319544 (k=4), A319545 (k=5), A319546 (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10). Sequence in context: A112333 A066868 A193900 * A143172 A004747 A155810 Adjacent sequences:  A319370 A319371 A319372 * A319374 A319375 A319376 KEYWORD sign,easy AUTHOR Wesley Ivan Hurt, Sep 17 2018 STATUS approved

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Last modified April 23 05:42 EDT 2021. Contains 343199 sequences. (Running on oeis4.)