A228958 a(n) = 1*2 + 3*4 + 5*6 + 7*8 + 9*10 + 11*12 + 13*14 + ... + (up to n).

1, 2, 5, 14, 19, 44, 51, 100, 109, 190, 201, 322, 335, 504, 519, 744, 761, 1050, 1069, 1430, 1451, 1892, 1915, 2444, 2469, 3094, 3121, 3850, 3879, 4720, 4751, 5712, 5745, 6834, 6869, 8094, 8131, 9500, 9539, 11060, 11101, 12782, 12825, 14674, 14719, 16744

Offset: 1

Author

Robert Pfister, Sep 09 2013

Keywords

• nonn
• easy

Comments

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=2. - Wesley Ivan Hurt, Sep 10 2018

a(2n) is the total area of the family of n rectangles, where the k-th rectangle has dimensions (2k) X (2k-1). - Wesley Ivan Hurt, Oct 01 2018

Examples

			1                            =   1
1*2                          =   2
1*2 + 3                      =   5
1*2 + 3*4                    =  14
1*2 + 3*4 + 5                =  19
1*2 + 3*4 + 5*6              =  44
1*2 + 3*4 + 5*6 + 7          =  51
1*2 + 3*4 + 5*6 + 7*8        = 100
1*2 + 3*4 + 5*6 + 7*8 + 9    = 109
1*2 + 3*4 + 5*6 + 7*8 + 9*10 = 190
...
		

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

Cf. A093361, (k=1) A000217, (k=2) this sequence, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

Programs

Magma

[(1/12)*(2*n^3+4*n+3/2+(3*n^2-6*n-3/2)*(-1)^n): n in [1..50]]; // Vincenzo Librandi, Sep 11 2018

Mathematica

a[n_?OddQ] := (2*n^3-3*n^2+10*n+3)/12; a[n_?EvenQ] := n*(n+2)*(2*n-1)/12; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Sep 10 2013 *)
CoefficientList[Series[x(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3), {x, 0, 40}], x] (* Stefano Spezia, Sep 23 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,2,5,14,19,44,51},50] (* Harvey P. Dale, Mar 11 2023 *)

PARI

Vec( x*(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3) + O(x^66) ) \\ Joerg Arndt, Sep 17 2013

Formula

a(n) = (1/12)*(2*n^3+4*n+3/2+(3*n^2-6*n-3/2)*(-1)^n). [based on Alcover program]

G.f.: x*(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3). [Joerg Arndt, Sep 13 2013]

E.g.f.: (x*(9 + 9*x + 2*x^2)*cosh(x) + (3 + 3*x + 3*x^2 + 2*x^3)*sinh(x))/12. - Stefano Spezia, Apr 18 2023

Extensions

Definition corrected by Ivan Panchenko, Dec 02 2013