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A130518
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a(n) = Sum_{k=0..n} floor(k/3). (Partial sums of A002264.)
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19
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0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570
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OFFSET
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0,5
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COMMENTS
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Complementary with A130481 regarding triangular numbers, in that A130481(n) + 3*a(n) = n(n+1)/2 = A000217(n).
Apart from offset, the same as A062781. - R. J. Mathar, Jun 13 2008
Apart from offset, the same as A001840. - Michael Somos, Sep 18 2010
The sum of any three consecutive terms is a triangular number. - J. M. Bergot, Nov 27 2014
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
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FORMULA
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G.f.: x^3 / ((1-x^3)*(1-x)^2).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = (1/2)*floor(n/3)*(2*n - 1 - 3*floor(n/3)))) = A002264(n)*(2n - 1 - 3*A002264(n))/2.
a(n) = (1/2)*A002264(n)*(n - 1 + A010872(n)).
a(n) = round(n*(n-1)/6) = round((n^2-n-1)/6) = floor(n*(n-1)/6) = ceiling((n+1)*(n-2)/6). - Mircea Merca, Nov 28 2010
a(n) = a(n-3) + n - 2, n > 2. - Mircea Merca, Nov 28 2010
a(n) = A214734(n, 1, 3). - Renzo Benedetti, Aug 27 2012
a(3n) = A000326(n), a(3n+1) = A005449(n), a(3n+2) = 3*A000217(n) = A045943(n). - Philippe Deléham, Mar 26 2013
a(n) = (3*n*(n-1) - (-1)^n*((1+i*sqrt(3))^(n-2) + (1-i*sqrt(3))^(n-2))/2^(n-3) - 2)/18, where i=sqrt(-1). - Bruno Berselli, Nov 30 2014
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MAPLE
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seq(floor(n*(n-1)/6), n=0..60); # Robert Israel, Nov 27 2014
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MATHEMATICA
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Table[n, {n, 0, 19}, {3}] // Flatten // Accumulate (* Jean-François Alcover, Jun 05 2013 *)
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PROG
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(Sage) [floor(binomial(n, 2)/3) for n in range(0, 60)] # Zerinvary Lajos, Dec 01 2009
(MAGMA) [Round(n*(n-1)/6): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011
(PARI) a(n)=n*(n-1)\/6 \\ Charles R Greathouse IV, Jun 05 2013
(GAP) List([0..60], n-> Int(n*(n-1)/6)); # G. C. Greubel, Aug 31 2019
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CROSSREFS
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Cf. A002265, A002266, A004526, A010872, A010873, A010874, A062781, A130482, A130483.
Sequence in context: A062781 A145919 A058937 * A001840 A022794 A025693
Adjacent sequences: A130515 A130516 A130517 * A130519 A130520 A130521
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KEYWORD
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nonn,easy
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AUTHOR
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Hieronymus Fischer, Jun 01 2007
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STATUS
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approved
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