A319867 a(n) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + ... + (up to the n-th term).

3, 6, 6, 12, 36, 126, 135, 198, 630, 642, 762, 1950, 1965, 2160, 4680, 4698, 4986, 9576, 9597, 9996, 17556, 17580, 18108, 29700, 29727, 30402, 47250, 47280, 48120, 71610, 71643, 72666, 104346, 104382, 105606, 147186, 147225, 148668, 202020, 202062, 203742

Offset

1,1

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = 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} (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, Sep 30 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)^5*(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 = 12;
a(5) = 3*2*1 + 6*5 = 36;
a(6) = 3*2*1 + 6*5*4 = 126;
a(7) = 3*2*1 + 6*5*4 + 9 = 135;
a(8) = 3*2*1 + 6*5*4 + 9*8 = 198;
a(9) = 3*2*1 + 6*5*4 + 9*8*7 = 630;
a(10) = 3*2*1 + 6*5*4 + 9*8*7 + 12 = 642;
a(11) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11 = 762;
a(12) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 = 1950;
a(13) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15 = 1965;
a(14) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14 = 2160;
a(15) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 = 4680;
a(16) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18 = 4698;
a(17) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18*17 = 4986;
a(18) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18*17*16 = 9576;
a(19) = 3*2*1 + 6*5*4 + 9*8*7 + 12*11*10 + 15*14*13 + 18*17*16 + 21 = 9597;
etc.

Maple

a:=(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((floor(j/k)-floor((j-1)/k))*(mul(j-i+1, i=1..k)), j=1..n): seq(a(n, 3), n=1..45); # Muniru A Asiru, Sep 30 2018

Mathematica

k:=3; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k]) * Product[n-i-j+k+1, {i, 1, j }], {j, 1, k-1} ] + Sum[(Floor[j/k]-Floor[(j-1)/k]) * Product[j-i+1, {i, 1, k}], {j, 1, n}]; Array[a, 50] (* Stefano Spezia, 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)^5*(1 + x + x^2)^4) + O(x^50)) \\ Colin Barker, Sep 30 2018

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), this sequence (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Cf. A268685 (trisection).

Sequence in context: A307204 A208445 A208796 * A202931 A240250 A239424

Adjacent sequences: A319864 A319865 A319866 * A319868 A319869 A319870

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Sep 29 2018

Status

approved