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 A173196 Partial sums of A002620. 14
 0, 0, 1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Essentially a duplicate of A002623: 0, 0, followed by A002623. The only primes in this sequence are 3, 7, and 13: for n > 2 both a(2*n+1) = n*(n+1)*(4*n+5)/6 and a(2*n) = n*(n+1)*(4*n-1)/6 are composite. - Bruno Berselli, Jan 19 2011 a(n-1) is the number of integer-sided scalene triangles with largest side <= n, including degenerate (i.e., collinear) triangles. a(n-2) is the number of non-degenerate integer-sided scalene triangles. - Alexander Evnin, Oct 12 2010 Also n-th differences of square pyramidal numbers (A000330) and numbers of triangles in triangular matchstick arrangement of side n (A002717). - Konstantin P. Lakov, Apr 13 2018 REFERENCES A. Yu. Evnin. Problem book on discrete mathematics. Moscow: Librokom, 2010; problem 787. (In Russian) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven. M. Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Int. Seq. 14 (2011) # 11.9.1. Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1). FORMULA G.f.: x^2 / ((1-x)^3 * (1-x^2)). a(n) = (4*n^3 + 6*n^2 - 4*n - 3 + 3*(-1)^n)/48. - Bruno Berselli, Jan 19 2011 a(n) = A002623(n-2) for n >= 2. - Martin von Gagern, Dec 05 2014 a(n) = Sum_{i=0..n} A002620(i) = Sum_{i=0..n} floor(i/2)*ceiling(i/2) = Sum_{i=0..n} floor(i^2/4). a(n) = round((2*n^3 + 3*n^2 - 2*n)/24) = round((4*n^3 + 6*n^2 - 4*n - 3)/48) = floor((2*n^3 + 3*n^2 - 2*n)/24) = ceiling((2*n^3 + 3*n^2 - 2*n - 3)/24). - Mircea Merca, Nov 23 2010 a(n) = a(n-2) + n*(n-1)/2, n > 1. - Mircea Merca, Nov 25 2010 a(n) = floor(n/2)*(floor(n/2)+1)*(8*ceiling(n/2) - 2*n - 1)/6. - Alexander Evnin, Oct 12 2010 a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jan 14 2021 E.g.f.: (x*(3 + 9*x + 2*x^2)*cosh(x) - (3 - 3*x - 9*x^2 - 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021 EXAMPLE a(57) = 0 + 0 + 1 + 2 + 4 + 6 + 9 + 12 + 16 + 20 + 25 + 30 + 36 + 42 + 49 + 56 + 64 + 72 + 81 + 90 + 100 + 110 + 121 + 132 + 144 + 156 + 169 + 182 + 196 + 210 + 225 + 240 + 256 + 272 + 289 + 306 + 324 + 342 + 361 + 380 + 400 + 420 + 441 + 462 + 484 + 506 + 529 + 552 + 576 + 600 + 625 + 650 + 676 + 702 + 729 + 756 + 784 + 812 = 15834. MATHEMATICA CoefficientList[Series[x^2/((1 - x)^3 (1 - x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *) Accumulate[Floor[Range[0, 60]^2/4]] (* or *) LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 1, 3, 7}, 60] (* Harvey P. Dale, Feb 09 2020 *) a[ n_] := Quotient[2 n^3 + 3 n^2 - 2 n, 24]; (* Michael Somos, Jan 14 2021 *) PROG (Magma) [Floor((2*n^3+3*n^2-2*n)/24): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011 CROSSREFS Cf. A000212, A000330, A002620, A002623, A002717, A002984, A007590, A024206, A056827, A072280, A087811, A118013, A118015. Sequence in context: A051336 A253896 A002623 * A081662 A091652 A334163 Adjacent sequences: A173193 A173194 A173195 * A173197 A173198 A173199 KEYWORD nonn,easy AUTHOR Jonathan Vos Post, Feb 12 2010 STATUS approved

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Last modified March 22 18:49 EDT 2023. Contains 361433 sequences. (Running on oeis4.)