A319866 a(n) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12*11 + ... + (up to the n-th term).

2, 2, 6, 14, 20, 44, 52, 100, 110, 190, 202, 322, 336, 504, 520, 744, 762, 1050, 1070, 1430, 1452, 1892, 1916, 2444, 2470, 3094, 3122, 3850, 3880, 4720, 4752, 5712, 5746, 6834, 6870, 8094, 8132, 9500, 9540, 11060, 11102, 12782, 12826, 14674, 14720, 16744

Offset: 1

Author

Wesley Ivan Hurt, Sep 29 2018

Keywords

• nonn
• easy

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=2.

The denominators of the generating functions for these sequences are (1 + x)*(1 - x^k)^(k+1). - Georg Fischer and Andrew Howroyd, Mar 07 2020

Examples

			a(1) = 2;
a(2) = 2*1 = 2;
a(3) = 2*1 + 4 = 6;
a(4) = 2*1 + 4*3 = 14;
a(5) = 2*1 + 4*3 + 6 = 20;
a(6) = 2*1 + 4*3 + 6*5 = 44;
a(7) = 2*1 + 4*3 + 6*5 + 8 = 52;
a(8) = 2*1 + 4*3 + 6*5 + 8*7 = 100;
a(9) = 2*1 + 4*3 + 6*5 + 8*7 + 10 = 110;
a(10) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 = 190;
a(11) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12 = 202;
etc.
		

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).

Crossrefs

For similar sequences, see: A000217 (k=1), this sequence (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Programs

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,2),n=1..50); # Muniru A Asiru, Sep 30 2018

Mathematica

k:=2; 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 *)
CoefficientList[Series[2/((-1 + x)^2 (1 + x)^2) + ( 2 (x + 3 x^3))/((-1 + x)^4 (1 + x)^3), {x, 0, 50}], x] (* Stefano Spezia, Sep 30 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{2,2,6,14,20,44,52},60] (* Harvey P. Dale, Mar 01 2025 *)

PARI

Vec(2*x*(1 - x^2 + 4*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^50)) \\ Colin Barker, Sep 30 2018

Formula

G.f.: 2*x/((-1 + x)^2*(1 + x)^2) + 2*(x^2 + 3*x^4)/((-1 + x)^4 (1 + x)^3). - Stefano Spezia, Sep 30 2018

From Colin Barker, Sep 30 2018: (Start)

a(n) = (4*n - 6*n + 3*n^2 + 2*n^3) / 12 for n even.

a(n) = (15 + 4*n + 6*n - 3*n^2 + 2*n^3) / 12 for n odd.

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)