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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).
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%I #22 Oct 05 2018 08:25:09

%S 3,6,6,0,-24,-114,-105,-42,390,378,258,-930,-915,-720,1800,1782,1494,

%T -3096,-3075,-2676,4884,4860,4332,-7260,-7233,-6558,10290,10260,9420,

%U -14070,-14037,-13014,18666,18630,17406,-24174,-24135,-22692,30660,30618,28938

%N 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).

%C 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.

%H Colin Barker, <a href="/A319886/b319886.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-4,4,0,-6,6,0,-4,4,0,-1,1).

%F From _Colin Barker_, Oct 01 2018: (Start)

%F 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).

%F 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.

%F (End)

%e a(1) = 3;

%e a(2) = 3*2 = 6;

%e a(3) = 3*2*1 = 6;

%e a(4) = 3*2*1 - 6 = 0;

%e a(5) = 3*2*1 - 6*5 = -24;

%e a(6) = 3*2*1 - 6*5*4 = -114;

%e a(7) = 3*2*1 - 6*5*4 + 9 = -105;

%e a(8) = 3*2*1 - 6*5*4 + 9*8 = -42;

%e a(9) = 3*2*1 - 6*5*4 + 9*8*7 = 390;

%e a(10) = 3*2*1 - 6*5*4 + 9*8*7 - 12 = 378;

%e a(11) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11 = 258;

%e a(12) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 = -930;

%e a(13) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15 = -915;

%e a(14) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14 = -720;

%e a(15) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 = 1800;

%e a(16) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18 = 1782;

%e a(17) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18*17 = 1494;

%e a(18) = 3*2*1 - 6*5*4 + 9*8*7 - 12*11*10 + 15*14*13 - 18*17*16 = -3096;

%e etc.

%p 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

%o (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

%Y 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).

%K sign,easy

%O 1,1

%A _Wesley Ivan Hurt_, Sep 30 2018