|
| |
|
|
A045944
|
|
Rhombic matchstick numbers: a(n) = n*(3*n+2).
|
|
69
|
|
|
|
0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008, 1121, 1240, 1365, 1496, 1633, 1776, 1925, 2080, 2241, 2408, 2581, 2760, 2945, 3136, 3333, 3536, 3745, 3960, 4181, 4408, 4641, 4880, 5125, 5376, 5633, 5896, 6165, 6440
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the n-th term of the sequence found by reading the line from 0 in the direction 0,5,.... The spiral begins:
.
85--84--83--82--81--80
. \
56--55--54--53--52 79
/ . \ \
57 33--32--31--30 51 78
/ / . \ \ \
58 34 16--15--14 29 50 77
/ / / . \ \ \ \
59 35 17 5---4 13 28 49 76
/ / / / . \ \ \ \ \
60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / /
61 37 19 7 1---2 11 26 47 74
\ \ \ \ / / / /
62 38 20 8---9--10 25 46 73
\ \ \ / / /
63 39 21--22--23--24 45 72
\ \ / /
64 40--41--42--43--44 71
\ /
65--66--67--68--69--70
(End)
Connection to triangular numbers: 4T_n + S_n where T_n is triangular number n and S_n is square number n. - William A. Tedeschi, Sep 12 2010
Also, second octagonal numbers. - Bruno Berselli, Jan 13 2011
Sequence found by reading the line from 0, in the direction 0, 16, ... and the line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
Let P denote the points from the n X n grid. A(n-1) also coincides with the minimum number of points Q needed to "block" P, that is, every line segment spanned by two points from P must contain one point from Q. - Manfred Scheucher, Aug 30 2018
Also the number of internal edges of an (n+1)*(n+1) "square" of hexagons; i.e., n+1 rows, each of n+1 edge-adjacent hexagons, stacked with minimal overhang. - Jon Hart, Sep 29 2019
|
|
|
LINKS
|
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Leo Tavares, Illustration: Square Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
|
|
|
FORMULA
|
O.g.f.: x*(5+x)/(1-x)^3. - R. J. Mathar, Jan 07 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=5, a(2)=16. - Harvey P. Dale, May 06 2011
a(n) = a(n-1) + 6*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
For n > 0, a(n)^3 + (a(n)+1)^3 + ... + (a(n)+n)^3 + 2*A000217(n)^2 = (a(n) + n + 1)^3 + ... + (a(n) + 2n)^3; see also A033954. - Charlie Marion, Dec 08 2007
a(n) = Sum_{i=0..n-1} A016969(i) for n > 0. - Bruno Berselli, Jan 13 2011
a(n) = A174709(6*n+4). - Philippe Deléham, Mar 26 2013
a(n) = A001082(2*n). - Michael Turniansky, Aug 24 2013
Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A002378(n) + A014105(n). - J. M. Bergot, Apr 24 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - Amiram Eldar, Jul 03 2020
E.g.f.: exp(x)*x*(5 + 3*x). - Stefano Spezia, Jun 08 2021
From Leo Tavares, Oct 14 2021: (Start)
a(n) = A000290(n) + 4*A000217(n). See Square Stars illustration.
a(n) = A000567(n+2) - A022144(n+1)
a(n) = A005563(n) + A001105(n).
a(n) = A056109(n) - 1. (End)
|
|
|
MATHEMATICA
|
Table[n*(3n+2), {n, 0, 60}] (* Harvey P. Dale, May 05 2011 *)
LinearRecurrence[{3, -3, 1}, {0, 5, 16}, 60] (* Harvey P. Dale, Jan 19 2016 *)
CoefficientList[Series[x*(5 + x)/(1 - x)^3, {x, 0, 60}], x] (* Stefano Spezia, Sep 01 2018 *)
|
|
|
PROG
|
(PARI) a(n)=n*(3*n+2) \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [n*(3*n+2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 24 2017
|
|
|
CROSSREFS
|
Bisection of A001859. See Comments of A135713.
Cf. A000217, A000567, A001082, A002378, A016969, A049450, A174709.
Cf. second n-gonal numbers: A005449, A014105, A147875, A179986, A033954, A062728, A135705.
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. A000290, A022144, A005563, A001105.
Cf. A056109.
Sequence in context: A063076 A270805 A132479 * A038361 A227719 A172166
Adjacent sequences: A045941 A045942 A045943 * A045945 A045946 A045947
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
R. K. Guy
|
|
|
STATUS
|
approved
|
| |
|
|
