A319373 a(n) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 - ... + (up to n).

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

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