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A228958 a(n) = 1*2 + 3*4 + 5*6 + 7*8 + 9*10 + 11*12 + 13*14 + ... + (up to n). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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).

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]

EXAMPLE

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

...

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-Fran├žois Alcover, Sep 10 2013 *)

CoefficientList[Series[(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3), {x, 0, 40}], x] (* Stefano Spezia, Sep 23 2018 *)

PROG

(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

(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

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.

Sequence in context: A191119 A089410 A118670 * A216669 A015633 A176191

Adjacent sequences:  A228955 A228956 A228957 * A228959 A228960 A228961

KEYWORD

nonn,easy

AUTHOR

Robert Pfister, Sep 09 2013

EXTENSIONS

Definition corrected by Ivan Panchenko, Dec 02 2013

STATUS

approved

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Last modified June 1 04:37 EDT 2020. Contains 334758 sequences. (Running on oeis4.)