Search: a304993 -id:a304993
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A045943
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Triangular matchstick numbers: a(n) = 3*n*(n+1)/2.
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+10
128
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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
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OFFSET
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0,2
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COMMENTS
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Also, 3 times triangular numbers, a(n) = 3*A000217(n).
In the 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 Elemer, Jun 20 2005
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 Janjic, Sep 19 2007
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. - Claudio Meller, Oct 08 2010
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. - John M. Campbell, Jul 17 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
A005449(a(n)) = A000332(3n + 3) = C(3n + 3, 4), a second pentagonal number of triangular matchstick number index number. Additionally, a(n) - 2n is a pentagonal number (A000326). - Raphie Frank, Dec 31 2012
Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 5376 or 17920 or 20160. - Philippe A.J.G. Chevalier, Dec 28 2015
Also the number of 4-cycles in the (n+4)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Number of terms less than 10^k, k=0,1,2,3,...: 1, 3, 8, 26, 82, 258, 816, 2582, 8165, 25820, 81650, 258199, 816497, 2581989, 8164966, ... - Muniru A Asiru, Jan 24 2018
Numbers of the form 3*m*(2*m + 1) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Column 1 of A273464. (Number of ways to select a unit lozenge inside an isosceles triangle of side length n; all vertices on a hexagonal lattice.) - R. J. Mathar, Jul 10 2019
Total number of pips in the n-th suit of a double-n domino set. - Ivan N. Ianakiev, Aug 23 2020
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 543.
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LINKS
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Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
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FORMULA
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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) = T(2*n) - T(n-1), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 06 2020
Sum_{n>=1} 1/a(n) = 2/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/3. (End)
Product_{n>=1} (1 - 1/a(n)) = -(3/(2*Pi))*cos(sqrt(11/3)*Pi/2). - Amiram Eldar, Feb 21 2023
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EXAMPLE
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T(n), the triangular numbers = number of nodes,
a(n-1) = number of edges in the T(n) graph:
o (T(1) = 1, a(0) = 0)
o
/ \ (T(2) = 3, a(1) = 3)
o - o
o
/ \
o - o (T(3) = 6, a(2) = 9)
/ \ / \
o - o - o
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MAPLE
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MATHEMATICA
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CoefficientList[Series[-3 x/(x - 1)^3, {x, 0, 47}], x] (* Robert G. Wilson v, Jan 29 2015 *)
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PROG
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(Common Lisp) (defun tri (i) (if (eq i 0) 0 (+ (* 3 (- i 1)) (tri (- i 1))))) // Stephen Balaban, Jul 25 2011
(Haskell) a n = sum [x | x <- [n..2*n]] -- Peter Kagey, Jul 27 2015
(GAP) List([0..10^4], n -> 3*n*(n+1)/2); # Muniru A Asiru, Jan 24 2018
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CROSSREFS
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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. A002378 (3-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
This sequence: Sum_{k = n..2*n} k.
Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.
Similar sequences are listed in A316466.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A050409
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Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.
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+10
13
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0, 5, 29, 86, 190, 355, 595, 924, 1356, 1905, 2585, 3410, 4394, 5551, 6895, 8440, 10200, 12189, 14421, 16910, 19670, 22715, 26059, 29716, 33700, 38025, 42705, 47754, 53186, 59015, 65255, 71920, 79024, 86581, 94605, 103110, 112110, 121619
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = n*(n+1)*(14*n+1)/6.
G.f.: x*(5+9*x)/(1-x)^4.
E.g.f.: x*(30 + 57*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019
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MAPLE
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 5, 29, 86}, 40] (* Vincenzo Librandi, Jun 22 2012 *)
Table[(n(n+1)(14n+1))/6, {n, 0, 40}] (* Harvey P. Dale, Mar 08 2020 *)
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PROG
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(Magma) [&+[k^2: k in [n..2*n]]: n in [0..40]]; // Bruno Berselli, Feb 11 2011
(Magma) I:=[0, 5, 29, 86]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012
(PARI) a(n)=sum(k=n, n+n, k^2)
(PARI) vector(40, n, n*(n-1)*(14*n-13)/6) \\ G. C. Greubel, Oct 30 2019
(Sage) [n*(n+1)*(14*n+1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019
(GAP) List([0..40], n-> n*(n+1)*(14*n+1)/6); # G. C. Greubel, Oct 30 2019
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CROSSREFS
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Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999
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STATUS
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approved
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A141433
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Triangle T(n, k) = (k-1)*(3*n-k), read by rows.
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+10
2
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0, 0, 4, 0, 7, 12, 0, 10, 18, 24, 0, 13, 24, 33, 40, 0, 16, 30, 42, 52, 60, 0, 19, 36, 51, 64, 75, 84, 0, 22, 42, 60, 76, 90, 102, 112, 0, 25, 48, 69, 88, 105, 120, 133, 144, 0, 28, 54, 78, 100, 120, 138, 154, 168, 180
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} Sum_{n>=0} T(n,k)*x^n*y^k = y^2*x*(2*x+1-3*y)/((1-y)^3*(x-1)^2). (G.f. for the full array, not just the triangular subspace) - R. J. Mathar, Feb 19 2020
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EXAMPLE
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Triangle begins as:
0;
0, 4;
0, 7, 12;
0, 10, 18, 24;
0, 13, 24, 33, 40;
0, 16, 30, 42, 52, 60;
0, 19, 36, 51, 64, 75, 84;
0, 22, 42, 60, 76, 90, 102, 112;
0, 25, 48, 69, 88, 105, 120, 133, 144;
0, 28, 54, 78, 100, 120, 138, 154, 168, 180;
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MAPLE
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A141433 := proc(n, m) (m-1)*(3*n-m) ; end proc:
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MATHEMATICA
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Flatten[Table[(m-1)(3n-m), {n, 10}, {m, n}]] (* Harvey P. Dale, Feb 04 2016 *)
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PROG
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(Magma) [(k-1)*(3*n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021
(Sage) flatten([[(k-1)*(3*n-k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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