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A130519
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Sum {0<=k<=n, floor(k/4)} (Partial sums of A002265).
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4
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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; internal format)
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OFFSET
| 0,6
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COMMENTS
| Complementary with A130482 regarding triangular numbers, in that A130482(n)+4*a(n)=n(n+1)/2 = A000217(n).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index to sequences with linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).
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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) [From Mitch Harris (maharri(AT)gmail.com), Sep 08 2008]
a(n) = round(n*(n-2)/8) = round((n^2-2*n-1)/8) = ceil((n+1)*(n-3)/8) [From Mircea Merca (mircea(AT)teacher.com), 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. [From Mircea Merca (mircea(AT)teacher.com), Nov 28 2010]
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
Euler transform of length 4 sequence [ 2, 0, 0, 1]. - Michael Somos, Oct 14 2011
a(2 - n) = a(n). - Michael Somos, Oct 14 2011
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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
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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
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MATHEMATICA
| a[ n_] := Quotient[ (n - 1)^2, 8] (* Michael Somos, Oct 14 2011 *)
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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
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CROSSREFS
| Cf. A002264, A002266, A004526, A010872, A010873, A010874, A130481, A130483.
Sequence in context: A056168 A054041 A019293 * A001972 A005705 A139542
Adjacent sequences: A130516 A130517 A130518 * A130520 A130521 A130522
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KEYWORD
| nonn,easy
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AUTHOR
| Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
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EXTENSIONS
| Partially edited by R. J. Mathar, Jul 11 2009
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