%I A014557
%S A014557 0,0,0,0,0,0,2,4,8,12,20,28,40,52,70,88,112,136,168,200,240,280,
%T A014557 330,380,440,500,572,644,728,812,910,1008,1120,1232,1360,1488,1632,
%U A014557 1776,1938,2100,2280,2460,2660,2860,3080,3300,3542,3784,4048,4312
%N A014557 Multiplicity of K_3 in K_n.
%C A014557 Comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 29 2006: 
               (Start)
%C A014557 n divides a(n) for n = {1,2,3,4,5,8,10,13,14,16,17,20,22,25,26,28,29,
               32,34,37,38,40,41,44,46,49,50,52,53,56,58,61,62,64,65,68,70,73,74,
               76,77,80,82,85,86,88,89,92,94,97,98,100,...}.
%C A014557 Prime p divides a(p) for p = {2,3,5,13,17,29,37,41,53,61,73,89,97,101,
               109,113,137,149,157,173,181,193,197,...} = (2,3) and all primes from 
               A002144(n) Pythagorean primes: primes of form 4n+1.
%C A014557 (n+1) divides a(n) for n = {1,2,3,4,5,19,27,43,51,67,75,91,99,...}.
%C A014557 (p+1) divides a(p) for prime p = {2,3,5,19,43,67,139,163,211,283,307,
               331,379,499,523,547,571,619,643,691,739,787,811,859,883,907,...} 
               = {2,5} and all primes from A107154(n) Primes of the form 3x^2+16y^2.
%C A014557 (n-1) divides a(n) for n = {2,3,4,5,21,29,45,53,69,77,93,101,...}.
%C A014557 (p-1) divides a(p) for prime p = {2,3,5,29,53,101,149,173,197,269,293,
               317,389,461,509,557,653,677,701,773,797,821,941,..} = {2,3} and all 
               primes from A107003(n) Primes of the form 5x^2+2xy+5y^2, with x and 
               y any integer.
%C A014557 (n-2) divides a(n) for n = {3,4,5,12,16,24,28,36,40,48,52,60,64,72,76,
               84,88,96,100,...} = {3,5} and 4*A032766(n) Numbers congruent to 0 
               or 1 mod 3.
%C A014557 (n+3) divides a(n) for n = {1,2,3,4,5,9,11,18,32,39}.
%C A014557 (n-3) divides a(n) for n = {4,5,7,9,23,31,47,55,71,79,95,103,119,127,
               143,151,167,175,...}.
%C A014557 (p+3) divides a(p) for prime p = {5,7,23,31,47,71,79,103,127,151,167,
               191,199,...} = [5} and all primes from A007522(n) Primes of form 
               8n+7.
%C A014557 (n-4) divides a(n) for n = {5,6,8,11,12,14,15,18,20,23,24,26,27,30,32,
               35,36,38,39,42,44,47,48,50,...}.
%C A014557 (p-4) divides a(p) for prime p = {5,11,23,47,59,71,83,107,131,167,179,
               191,...} = {5} and all primes from A068231(n) Primes congruent to 
               11 (mod 12).
%C A014557 (n+5) divides a(n) for n = {1,2,3,4,5,30,31,45,58,145}.
%C A014557 (n-5) divides a(n) for n = {6,7,9,10,20,25,33,49,57,73,81,97,105,...}.
%C A014557 (p-5) divides a(p) for prime p = {7,73,97,193,241,313,337,409,433,457,
               577,601,673,769,937,...} = {7} and all primes from A107008(n) Primes 
               of the form x^2+24y^2. (End)
%D A014557 Goodman, A. W., On Sets of Acquaintances and Strangers at Any Party, 
               Amer. Math. Monthly 66, 778-783, 1959.
%D A014557 V. Vijayalakshmi, Multiplicity of triangles in cocktail party graphs, 
               Discrete Math., 206 (1999), 217-218.
%H A014557 Alexander Adamchuk, <a href="b014557.txt">Table of n, a(n) for n = 0..100</
               a>
%H A014557 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ExtremalGraph.html">Link to a section of The World of Mathematics.</
               a>
%H A014557 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MonochromaticForcedTriangle.html">Link to a section of The World 
               of Mathematics.</a>
%F A014557 a(n) = binomial[n,3] - Floor[n/2*Floor[((n-1)/2)^2]]. - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Nov 29 2006
%p A014557 A049322 := proc(n) local u; if n mod 2 = 0 then u := n/2; RETURN(u*(u-1)*(u-2)/
               3); elif n mod 4 = 1 then u := (n-1)/4; RETURN(u*(u-1)*(4*u+1)*2/
               3); else u := (n-3)/4; RETURN(u*(u+1)*(4*u-1)*2/3); fi; end;
%t A014557 Table[Binomial[n,3] - Floor[n/2*Floor[((n-1)/2)^2]],{n,0,100}] - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Nov 29 2006
%Y A014557 Twice A008804.
%Y A014557 Cf. A002144, A107154, A107003, A032766, A007522, A068231, A107008.
%Y A014557 Sequence in context: A059793 A118029 A049322 this_sequence A023598 A103258 
               A100684
%Y A014557 Adjacent sequences: A014554 A014555 A014556 this_sequence A014558 A014559 
               A014560
%K A014557 nonn,nice,easy
%O A014557 0,7
%A A014557 Eric Weisstein (eric(AT)weisstein.com)
%E A014557 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Mar 22, 
               2004.