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A045943
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Triangular matchstick numbers: 3*n*(n+1)/2.
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62
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0, 3, 9, 18, 30, 45, 63, 84, 108, 135, 165, 198, 234, 273, 315, 360, 408, 459, 513, 570, 630, 693, 759, 828, 900, 975, 1053, 1134, 1218, 1305, 1395, 1488, 1584, 1683, 1785, 1890, 1998, 2109, 2223, 2340, 2460, 2583, 2709, 2838, 2970, 3105, 3243, 3384, 3528
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
a(n) = A126890(n+1,n-1) for n>1. - Reinhard Zumkeller, Dec 30 2006
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
a(n) + A145919(3n+3) = 0. [From Matthew Vandermast, Oct 28 2008]
a(n) = A000217(2*n) - A000217(n-1); A179213(n) <= a(n). [From Reinhard Zumkeller, Jul 05 2010]
a(n) =is also the smallest number that may be written both as the sum of n-1 consecutive positive integers and n consecutive positive integers [From Claudio L Meller, Oct 08 2010]
Also, 3 times triangular numbers (see Crossrefs). [R. K. Guy]
For n >= 3, a(n) equals 4^(2+n)*pi^(1 - n) times the coefficient of zeta(3) in the following integral with upper bound pi/4 and lower bound 0: int x^(n+1) tan x dx. [From John M. Campbell, Jul 17, 2011]
Intuitively, add a triangle of three lines branching from each of the last nodes on the bottom of the graph, thus, each iteration adds 3 * (the number of nodes on the bottom of the last iteration <==> n). - Stephen Balaban, Jul 25 2011
Sequence found by reading the line from 0, in the direction 0, 3,... and the same line from 0, in the direction 0, 9,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. This is one of the orthogonal axes of the spiral. The other is A032528. - Omar E. Pol, Sep 08 2011
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REFERENCES
| 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.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Milan Janjic, Two Enumerative Functions
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| 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, Jan 15 2004
a(n)=a(n-1)+3*n (with a(0)=0) [From Vincenzo Librandi, Nov 18 2010]
G.f.: 3*x/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*A000217(n). - Omar E. Pol, Sep 17 2011
a(n) = A005448(n+1) - 1. Omar E. Pol, Oct 03 2011
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EXAMPLE
| T(n), The Triangular Numbers
a(n), The Triangular Connecting Numbers
(T(0) = 0, a(0) = 0)
o (T(1) = 1, a(1) = 0)
o
/ \ (T(2) = 3, a(2) = 3)
o - o
o
/ \
o - o (T(3) = 6, a(3) = 9)
/ \ / \
o - o - o
... - Stephen Balaban, Jul 25 2011
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MAPLE
| [seq(3*binomial(n, 2), n=1..49)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2006
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MATHEMATICA
| Table[3*n*(n+1)/2, {n, 0, 50}] [From Vladimir Orlovsky, Oct 31 2008]
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PROG
| (CLISP) (defun tri (i) (if (eq i 0) 0 (+ (* 3 (- i 1)) (tri (- i 1))))) // Stephen Balaban, Jul 25 2011
(MAGMA) [3*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, May 02 2011
(PARI) a(n)=3*binomial(n+1, 2) \\ Charles R Greathouse IV, Jun 16 2011
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CROSSREFS
| Cf. A005448, A002378, A046092.
3 times n-gonal numbers: A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
Cf. A051162.
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.
Cf. A126804. [From Reinhard Zumkeller, Jul 05 2010]
Cf. A001318, A032528. - Omar E. Pol, Sep
Sequence in context: A100967 A193567 A134479 * A184969 A194113 A194114
Adjacent sequences: A045940 A045941 A045942 * A045944 A045945 A045946
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KEYWORD
| nonn,easy
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AUTHOR
| R. K. Guy
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