A319886 a(n) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18*17*16 + ... - (up to the n-th term).

3, 6, 6, 0, -24, -114, -105, -42, 390, 378, 258, -930, -915, -720, 1800, 1782, 1494, -3096, -3075, -2676, 4884, 4860, 4332, -7260, -7233, -6558, 10290, 10260, 9420, -14070, -14037, -13014, 18666, 18630, 17406, -24174, -24135, -22692, 30660, 30618, 28938

Offset

1,1

Comments

For similar sequences that alternate in descending blocks of k natural numbers, we have: a(n) = (-1)^floor(n/k) * Sum_{j=1..k-1} (floor((n-j)/k) - floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (-1)^(floor(j/k)+1) * (floor(j/k) - floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=3.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,-4,4,0,-6,6,0,-4,4,0,-1,1).

Formula

From Colin Barker, Oct 01 2018: (Start)

G.f.: 3*x*(1 + x + 2*x^3 - 4*x^4 - 30*x^5 + x^6 - 5*x^7 + 24*x^8) / ((1 - x)*(1 + x)^4*(1 - x + x^2)^4).

a(n) = a(n-1) - 4*a(n-3) + 4*a(n-4) - 6*a(n-6) + 6*a(n-7) - 4*a(n-9) + 4*a(n-10) - a(n-12) + a(n-13) for n>13.

(End)

Example

   a(1) = 3;
   a(2) = 3*2 = 6;
   a(3) = 3*2*1 = 6;
   a(4) = 3*2*1 - 6 = 0;
   a(5) = 3*2*1 - 6*5 = -24;
   a(6) = 3*2*1 - 6*5*4 = -114;
   a(7) = 3*2*1 - 6*5*4 + 9 = -105;
   a(8) = 3*2*1 - 6*5*4 + 9*8 = -42;
   a(9) = 3*2*1 - 6*5*4 + 9*8*7 = 390;
  a(10) = 3*2*1 - 6*5*4 + 9*8*7 - 12 = 378;
  a(11) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11 = 258;
  a(12) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 = -930;
  a(13) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15 = -915;
  a(14) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14 = -720;
  a(15) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 = 1800;
  a(16) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18 = 1782;
  a(17) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18*17 = 1494;
  a(18) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18*17*16 = -3096;
  etc.

Maple

a:=(n, k)->(-1)^(floor(n/k))* add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1, i=1..j)), j=1..k-1) + add( (-1)^(floor(j/k)+1)*(floor(j/k)-floor((j-1)/k))*(mul(j-i+1, i=1..k)), j=1..n): seq(a(n, 3), n=1..50); # Muniru A Asiru, Sep 30 2018

Prog

(PARI) Vec(3*x*(1 + x + 2*x^3 - 4*x^4 - 30*x^5 + x^6 - 5*x^7 + 24*x^8) / ((1 - x)*(1 + x)^4*(1 - x + x^2)^4) + O(x^50)) \\ Colin Barker, Oct 01 2018

Crossrefs

For similar sequences, see: A001057 (k=1), A319885 (k=2), this sequence (k=3), A319887 (k=4), A319888 (k=5), A319889 (k=6), A319890 (k=7), A319891 (k=8), A319892 (k=9), A319893 (k=10).

Sequence in context: A256460 A152139 A285628 * A021736 A091478 A239567

Adjacent sequences: A319883 A319884 A319885 * A319887 A319888 A319889

Keyword

sign,easy

Author

Wesley Ivan Hurt, Sep 30 2018

Status

approved