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A173196 Partial sums of A002620. 13
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) [Alexander Evnin, Oct 12 2010]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

W. Lanssens, B. Demoen, 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) = ceil((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

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 *)

PROG

(MAGMA) [Floor((2*n^3+3*n^2-2*n)/24): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

CROSSREFS

Cf. A000212, A002620, A002623, A002984, A007590, A024206, A056827, A072280, A087811, A118013, A118015.

Sequence in context: A051336 A253896 A002623 * A081662 A091652 A291546

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 August 18 12:35 EDT 2018. Contains 313832 sequences. (Running on oeis4.)