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A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.) 16
0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Complementary to A130482 with respect to triangular numbers, in that A130482(n) + 4*a(n) = n(n+1)/2 = A000217(n).

Disregarding the first three 0's the resulting sequence a'(n) is the sum of the positive integers <= n that have the same residue modulo 4 as n. This is the additive counterpart of the quadruple factorial numbers. - Peter Luschny, Jul 06 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

FORMULA

G.f.: x^4/((1-x^4)*(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).

a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-4) -2*a(n-5) +1*a(n-6).

a(n) = floor(n/4)*(n - 1 - 2*floor(n/4)) = A002265(n)*(n - 1 - 2*A002265(n)).

a(n) = (1/2)*A002265(n)*(n - 2 + A010873(n)).

a(n) = floor((n-1)^2/8). - Mitch Harris, Sep 08 2008

a(n) = round(n*(n-2)/8) = round((n^2-2*n-1)/8) = ceiling((n+1)*(n-3)/8). - Mircea Merca, Nov 28 2010

a(n) = A001972(n-4), n>3. - Franklin T. Adams-Watters, Jul 10 2009

a(n) = a(n-4)+n-3, n>3. - Mircea Merca, Nov 28 2010

Euler transform of length 4 sequence [ 2, 0, 0, 1]. - Michael Somos, Oct 14 2011

a(n) = a(2-n) for all n in Z. - Michael Somos, Oct 14 2011

a(n) = A214734(n, 1, 4). - Renzo Benedetti, Aug 27 2012

a(4n) = A000384(n), a(4n+1) = A001105(n), a(4n+2) = A014105(n), a(4n+3) = A046092(n). - Philippe Deléham, Mar 26 2013

a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - Wesley Ivan Hurt, Jan 23 2014

a(n) = ( 2*n^2-4*n-1+(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4)) )/16 = ( 2*n*(n-2) - (1-(-1)^n)*(1-2*i^(n*(n-1))) )/16, where i=sqrt(-1). - Luce ETIENNE, Aug 29 2014

EXAMPLE

x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 12*x^11 + ...

[ n] a(n)

---------

[ 4] 1

[ 5] 2

[ 6] 3

[ 7] 4

[ 8] 1 + 5

[ 9] 2 + 6

[10] 3 + 7

[11] 4 + 8

MAPLE

quadsum := n -> add(k, k = select(k -> k mod 4 = n mod 4, [$1 .. n])):

A130519 := n ->`if`(n<3, 0, quadsum(n-3)); seq(A130519(n), n=0..58); # Peter Luschny, Jul 06 2011

MATHEMATICA

a[ n_] := Quotient[ (n - 1)^2, 8]; (* Michael Somos, Oct 14 2011 *)

LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 0, 0, 1, 2}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

PROG

(PARI) {a(n) = (n - 1)^2 \ 8}; /* Michael Somos, Oct 14 2011 */

(MAGMA) [Round(n*(n-2)/8): n in [0..70]]; // Vincenzo Librandi, Jun 25 2011

CROSSREFS

Cf. A002264, A002265, A002266, A004526, A010872, A010873, A010874, A130481, A130483.

Sequence in context: A056168 A054041 A019293 * A001972 A005705 A139542

Adjacent sequences:  A130516 A130517 A130518 * A130520 A130521 A130522

KEYWORD

nonn,easy

AUTHOR

Hieronymus Fischer, Jun 01 2007

EXTENSIONS

Partially edited by R. J. Mathar, Jul 11 2009

STATUS

approved

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Last modified March 28 22:27 EDT 2017. Contains 284249 sequences.