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A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

%I A130519 #120 Aug 13 2022 06:22:58
%S A130519 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 A130519 78,84,91,98,105,112,120,128,136,144,153,162,171,180,190,200,210,220,
%U A130519 231,242,253,264,276,288,300,312,325,338,351,364,378,392,406,420,435,450
%N A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)
%C A130519 Complementary to A130482 with respect to triangular numbers, in that A130482(n) + 4*a(n) = n(n+1)/2 = A000217(n).
%C A130519 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 A130519 From _Heinrich Ludwig_, Dec 23 2017: (Start)
%C A130519 Column sums of (shift of rows = 4):
%C A130519   1 2 3 4 5 6  7  8  9 10 11 12 13 14 ...
%C A130519           1 2  3  4  5  6  7  8  9 10 ...
%C A130519                      1  2  3  4  5  6 ...
%C A130519                                  1  2 ...
%C A130519   .......................................
%C A130519   ---------------------------------------
%C A130519   1 2 3 4 6 8 10 12 15 18 21 24 28 32 ...
%C A130519   shift of rows = 1 see A000217
%C A130519   shift of rows = 2 see A002620
%C A130519   shift of rows = 3 see A001840
%C A130519   shift of rows = 5 see A130520
%C A130519 (End)
%C A130519 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 A130519 Vincenzo Librandi, <a href="/A130519/b130519.txt">Table of n, a(n) for n = 0..10000</a>
%H A130519 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 A130519 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 A130519 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 A130519 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A130519 G.f.: x^4/((1-x^4)*(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).
%F A130519 a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-4) -2*a(n-5) +1*a(n-6).
%F A130519 a(n) = floor(n/4)*(n - 1 - 2*floor(n/4)) = A002265(n)*(n - 1 - 2*A002265(n)).
%F A130519 a(n) = (1/2)*A002265(n)*(n - 2 + A010873(n)).
%F A130519 a(n) = floor((n-1)^2/8). - _Mitch Harris_, Sep 08 2008
%F A130519 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 A130519 a(n) = A001972(n-4), n>3. - _Franklin T. Adams-Watters_, Jul 10 2009
%F A130519 a(n) = a(n-4)+n-3, n>3. - _Mircea Merca_, Nov 28 2010
%F A130519 Euler transform of length 4 sequence [ 2, 0, 0, 1]. - _Michael Somos_, Oct 14 2011
%F A130519 a(n) = a(2-n) for all n in Z. - _Michael Somos_, Oct 14 2011
%F A130519 a(n) = A214734(n, 1, 4). - _Renzo Benedetti_, Aug 27 2012
%F A130519 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 A130519 a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - _Wesley Ivan Hurt_, Jan 23 2014
%F A130519 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 A130519 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 A130519 a(n) = (A002620(n-1) - A011765(n+1)) / 2, for n > 0. - _Yuchun Ji_, Feb 05 2021
%F A130519 Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - _Amiram Eldar_, Aug 13 2022
%e A130519 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 A130519 [ n] a(n)
%e A130519 ---------
%e A130519 [ 4] 1
%e A130519 [ 5] 2
%e A130519 [ 6] 3
%e A130519 [ 7] 4
%e A130519 [ 8] 1 + 5
%e A130519 [ 9] 2 + 6
%e A130519 [10] 3 + 7
%e A130519 [11] 4 + 8
%p A130519 quadsum := n -> add(k, k = select(k -> k mod 4 = n mod 4, [$1 .. n])):
%p A130519 A130519 := n ->`if`(n<3,0,quadsum(n-3)); seq(A130519(n),n=0..58); # _Peter Luschny_, Jul 06 2011
%t A130519 a[ n_] := Quotient[ (n - 1)^2, 8]; (* _Michael Somos_, Oct 14 2011 *)
%o A130519 (PARI) {a(n) = (n - 1)^2 \ 8}; /* _Michael Somos_, Oct 14 2011 */
%o A130519 (Magma) [Round(n*(n-2)/8): n in [0..70]]; // _Vincenzo Librandi_, Jun 25 2011
%o A130519 (Maxima) makelist(floor((n-1)^2/8), n, 0, 70); /* _Stefano Spezia_, Jan 04 2019 */
%o A130519 (GAP) a:=List([0..65],n->Sum([0..n],k->Int(k/4)));; Print(a); # _Muniru A Asiru_, Jan 04 2019
%o A130519 (Python)
%o A130519 def A130519(n): return (n-1)**2>>3  # _Chai Wah Wu_, Jul 30 2022
%Y A130519 Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520.
%Y A130519 Cf. A000290, A007590, A000212, A118015, A056827, A056834, A056838, A056865.
%K A130519 nonn,easy
%O A130519 0,6
%A A130519 _Hieronymus Fischer_, Jun 01 2007
%E A130519 Partially edited by _R. J. Mathar_, Jul 11 2009