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 A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.) 25
 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 From Heinrich Ludwig, Dec 23 2017: (Start) Column sums of (shift of rows = 4): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 1 2 3 4 5 6 7 8 9 10 ... 1 2 3 4 5 6 ... 1 2 ... ....................................... --------------------------------------- 1 2 3 4 6 8 10 12 15 18 21 24 28 32 ... shift of rows = 1 see A000217 shift of rows = 2 see A002620 shift of rows = 3 see A001840 shift of rows = 5 see A130520 (End) 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4. Darren Glass and Joshua Wagner, Arithmetical Structures on Paths With a Doubled Edge, arXiv:1903.01398 [math.CO], 2019. Nathan Pflueger, On non-primitive Weierstrass points, Alg. Number Th. 12 (2018) 1923-1947 and arXiv:1608.0566 [math.AG], 2016. 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 E.g.f.: (1/8)*((- 1 + x)*x*cosh(x) + 2*sin(x) + (- 1 - x + x^2)*sinh(x)). - Stefano Spezia, Jan 15 2019 a(n) = (A002620(n-1) - A011765(n+1)) / 2, for n > 0. - Yuchun Ji, Feb 05 2021 Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - Amiram Eldar, Aug 13 2022 EXAMPLE 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 + ... [ 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 *) 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 (Maxima) makelist(floor((n-1)^2/8), n, 0, 70); /* Stefano Spezia, Jan 04 2019 */ (GAP) a:=List([0..65], n->Sum([0..n], k->Int(k/4)));; Print(a); # Muniru A Asiru, Jan 04 2019 (Python) def A130519(n): return (n-1)**2>>3 # Chai Wah Wu, Jul 30 2022 CROSSREFS Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520. Cf. A000290, A007590, A000212, A118015, A056827, A056834, A056838, A056865. Sequence in context: A056168 A054041 A019293 * A001972 A328325 A005705 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 December 7 21:53 EST 2022. Contains 358670 sequences. (Running on oeis4.)