%I A045943
%S A045943 0,3,9,18,30,45,63,84,108,135,165,198,234,273,315,360,408,459,513,570,
%T A045943 630,693,759,828,900,975,1053,1134,1218,1305,1395,1488,1584,1683,1785,
%U A045943 1890,1998,2109,2223,2340,2460,2583,2709,2838,2970,3105,3243,3384,3528
%N A045943 Triangular matchstick numbers: 3n(n+1)/2.
%C A045943 In 24-bit RGB color cube, the number of color-lattice-points in r+g+b 
               = n planes at n < 256 equals the triangular numbers. For n = 256, 
               ..., 765 the number of legitimate color partitions is less than A000217(n) 
               because {r,g,b} components cannot exceed 255. For n=256,..,511, the 
               number of non-color partitions are computable with A045943(n-255), 
               while for n = 512-765, the number of color points in r+g+b planes 
               equals A000217(765-n). - Labos E. (labos(AT)ana.sote.hu), Jun 20 
               2005
%C A045943 a(n) = A126890(n+1,n-1) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 30 2006
%C A045943 If a 3-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then 
               a(n-3) is the number of 3-subsets of X intersecting both Y and Z. 
               - Milan R. Janjic (agnus(AT)blic.net), Sep 19 2007
%C A045943 a(n) + A145919(3n+3) = 0. [From Matthew Vandermast (ghodges14(AT)comcast.net), 
               Oct 28 2008]
%C A045943 Except for the first term, a(n)=3*n+a(n-1), (with a(1)=3) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]
%D A045943 Labos E.: On the number of RGB-colors we can distinguish. Partition Spectra. 
               Lecture at 7th Hungarian Conference on Biometry and Biomathematics. 
               Budapest. Jul 06, 2005.
%H A045943 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A045943 Alfred Hoehn, <a href="a000326.jpg">Illustration of initial terms of 
               A000326, A005449, A045943, A115067</a>
%F A045943 a(n) is the sum of n+1 integers starting from n, i.e. 1+2, 2+3+4, 3+4+5+6, 
               4+5+6+7+8, etc. - Jon Perry (perry(AT)globalnet.co.uk), Jan 15 2004
%F A045943 a(n)=3*n+a(n-1)-3 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A045943 For n=2, a(2)=3*2+0-3=3; n=3, a(3)=3*3+3-3=9; n=4, a(4)=3*4+9-3=18 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A045943 [seq(3*binomial(n,2),n=1..49)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 24 2006
%p A045943 a:=n->sum(3*j,j=0..n): seq(a(n), n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 04 2007
%p A045943 a:=n->sum(n+j, j=1..n)+n: seq(a(n), n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 29 2007
%p A045943 a:=n->sum(n, k=0..n):seq(a(n)+sum(k, k=1..n), n=0..48); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 10 2008
%p A045943 with(finance):seq(add(cashflows([n, k, 0], 0 ), k=0..n), n=0..45); # 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 22 2008]
%t A045943 s=0;lst={s};Do[s+=n;s+=n+1;s+=n+2;AppendTo[lst, s], {n, 0, 6!, 1}];lst 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 31 2008]
%t A045943 s = 0; lst = {}; Do[s += n; AppendTo[lst, s], {n, 0, 160, 3}]; lst [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
%t A045943 Table[Sum[i + n - 3, {i, 3, n}], {n, 2, 50}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 11 2009]
%Y A045943 3 times triangular numbers (A000217). Cf. A005448, A002378, A046092.
%Y A045943 Cf. A051162.
%Y A045943 The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 
               12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, 
               A059845, A140672, A140673, A140674, A140675, A151542.
%Y A045943 Sequence in context: A159794 A100967 A134479 this_sequence A127759 A064843 
               A093446
%Y A045943 Adjacent sequences: A045940 A045941 A045942 this_sequence A045944 A045945 
               A045946
%K A045943 nonn
%O A045943 0,2
%A A045943 R. K. Guy