%I #120 Aug 13 2022 06:22:58

%S 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,

%T 78,84,91,98,105,112,120,128,136,144,153,162,171,180,190,200,210,220,

%U 231,242,253,264,276,288,300,312,325,338,351,364,378,392,406,420,435,450

%N a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

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

%C 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

%C From _Heinrich Ludwig_, Dec 23 2017: (Start)

%C Column sums of (shift of rows = 4):

%C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...

%C 1 2 3 4 5 6 7 8 9 10 ...

%C 1 2 3 4 5 6 ...

%C 1 2 ...

%C .......................................

%C ---------------------------------------

%C 1 2 3 4 6 8 10 12 15 18 21 24 28 32 ...

%C shift of rows = 1 see A000217

%C shift of rows = 2 see A002620

%C shift of rows = 3 see A001840

%C shift of rows = 5 see A130520

%C (End)

%C Conjecture: a(n+2) is the maximum effective weight of a numerical semigroup S of genus n (see Nathan Pflueger). - _Stefano Spezia_, Jan 04 2019

%H Vincenzo Librandi, <a href="/A130519/b130519.txt">Table of n, a(n) for n = 0..10000</a>

%H Bakir Farhi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Farhi/farhi7.html">On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a)</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.

%H Darren Glass and Joshua Wagner, <a href="https://arxiv.org/abs/1903.01398">Arithmetical Structures on Paths With a Doubled Edge</a>, arXiv:1903.01398 [math.CO], 2019.

%H Nathan Pflueger, <a href="https://arxiv.org/abs/1608.05666">On non-primitive Weierstrass points</a>, Alg. Number Th. 12 (2018) 1923-1947 and arXiv:1608.0566 [math.AG], 2016.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).

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

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

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

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

%F a(n) = floor((n-1)^2/8). - _Mitch Harris_, Sep 08 2008

%F 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

%F a(n) = A001972(n-4), n>3. - _Franklin T. Adams-Watters_, Jul 10 2009

%F a(n) = a(n-4)+n-3, n>3. - _Mircea Merca_, Nov 28 2010

%F Euler transform of length 4 sequence [ 2, 0, 0, 1]. - _Michael Somos_, Oct 14 2011

%F a(n) = a(2-n) for all n in Z. - _Michael Somos_, Oct 14 2011

%F a(n) = A214734(n, 1, 4). - _Renzo Benedetti_, Aug 27 2012

%F a(4n) = A000384(n), a(4n+1) = A001105(n), a(4n+2) = A014105(n), a(4n+3) = A046092(n). - _Philippe Deléham_, Mar 26 2013

%F a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - _Wesley Ivan Hurt_, Jan 23 2014

%F 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

%F E.g.f.: (1/8)*((- 1 + x)*x*cosh(x) + 2*sin(x) + (- 1 - x + x^2)*sinh(x)). - _Stefano Spezia_, Jan 15 2019

%F a(n) = (A002620(n-1) - A011765(n+1)) / 2, for n > 0. - _Yuchun Ji_, Feb 05 2021

%F Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - _Amiram Eldar_, Aug 13 2022

%e G.f. = x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 12*x^11 + ...

%e [ n] a(n)

%e ---------

%e [ 4] 1

%e [ 5] 2

%e [ 6] 3

%e [ 7] 4

%e [ 8] 1 + 5

%e [ 9] 2 + 6

%e [10] 3 + 7

%e [11] 4 + 8

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

%p A130519 := n ->`if`(n<3,0,quadsum(n-3)); seq(A130519(n),n=0..58); # _Peter Luschny_, Jul 06 2011

%t a[ n_] := Quotient[ (n - 1)^2, 8]; (* _Michael Somos_, Oct 14 2011 *)

%o (PARI) {a(n) = (n - 1)^2 \ 8}; /* _Michael Somos_, Oct 14 2011 */

%o (Magma) [Round(n*(n-2)/8): n in [0..70]]; // _Vincenzo Librandi_, Jun 25 2011

%o (Maxima) makelist(floor((n-1)^2/8), n, 0, 70); /* _Stefano Spezia_, Jan 04 2019 */

%o (GAP) a:=List([0..65],n->Sum([0..n],k->Int(k/4)));; Print(a); # _Muniru A Asiru_, Jan 04 2019

%o (Python)

%o def A130519(n): return (n-1)**2>>3 # _Chai Wah Wu_, Jul 30 2022

%Y Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520.

%Y Cf. A000290, A007590, A000212, A118015, A056827, A056834, A056838, A056865.

%K nonn,easy

%O 0,6

%A _Hieronymus Fischer_, Jun 01 2007

%E Partially edited by _R. J. Mathar_, Jul 11 2009