|
| |
|
|
A056105
|
|
First spoke of a hexagonal spiral.
|
|
38
|
|
|
|
1, 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, 342, 409, 482, 561, 646, 737, 834, 937, 1046, 1161, 1282, 1409, 1542, 1681, 1826, 1977, 2134, 2297, 2466, 2641, 2822, 3009, 3202, 3401, 3606, 3817, 4034, 4257, 4486, 4721, 4962, 5209, 5462, 5721, 5986, 6257
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
Table of n, a(n) for n=0..46.
H. Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
|
|
|
FORMULA
|
a(n) = 3n^2 - 2n + 1 = a(n-1) + 6n - 5 = 2a(n-1) - a(n-2) + 6 = 3a(n-1) - 3a(n-2) + a(n-3) = A056106(n) - n = A056107(n) - 2n = A056108(n) - 3n = A056109(n) - 4n = A003215(n) - 5n.
A008810(3n-1) = A056109(-n) = a(n). - Michael Somos, Aug 03 2006.
a(n) = 6*n + a(n-1) - 5 (with a(0)=1). - Vincenzo Librandi, Aug 07 2010
G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
From Robert G. Wilson v, Jul 05 2014: (Start)
Each of the 6 primary spokes or rays has a generating formula as stated here:
1st: 90º A056105 3n^2 - 2n + 1
2nd: 30º A056106 3n^2 - n + 1
3rd: 330º A056107 3n^2 + 1
4th: 270º A056108 3n^2 + n + 1
5th: 210º A056109 3n^2 + 2n + 1
6th: 150º A003215 3n^2 + 3n + 1
Each of the 6 secondary spokes or rays has a generating formula as stated here:
1st: 60º 12n^2 - 27n + 16
2nd: 360º 12n^2 - 25n + 14
3rd: 300º 12n^2 - 23n + 12
4th: 240º 12n^2 - 21n + 10
5th: 180º 12n^2 - 19n + 8
6th: 120º 12n^2 - 17n + 6 = A033577(n+1)
(End)
a(n) = 1 + A000567(n). - Omar E. Pol, Apr 26 2017
|
|
|
EXAMPLE
|
a(1) = 6*1 + 1 - 5 = 2;
a(2) = 6*2 + 2 - 5 = 9;
a(3) = 6*3 + 9 - 5 = 22. - Vincenzo Librandi, Aug 07 2010
From Robert G. Wilson v, Jul 05 2014: (Start)
.
............363.362.361.360.359.358.357.356.355.354
.
..........301.300.299.298.297.296.295.294.293.292.291
.
........302.244.243.242.241.240.239.238.237.236.235.290
.
......303.245.193.192.191.190.189.188.187.186.185.234.289
.
....304.246.194.148.147.146.145.144.143.142.141.184.233.288
.
..305.247.195.149.109.108.107.106.105.104.103.140.183.232.287
.
306.248.196.150.110..76..75..74..73..72..71.102.139.182.231.286
.
..249.197.151.111..77..49..48..47..46..45..70.101.138.181.230.285
.
250.198.152.112..78..50..28..27..26..25..44..69.100.137.180.229.
.
..199.153.113..79..51..29..13..12..11..24..43..68..99.136.179.228
.
200.154.114..80..52..30..14...4...3..10..23..42..67..98.135.178.
.
..155.115..81..53..31..15...5...1...2...9..22..41..66..97.134.177
.
202.156.116..82..54..32..16...6...7...8..21..40..65..96.133.176.
.
..203.157.117..83..55..33..17..18..19..20..39..64..95.132.175.224
.
256.204.158.118..84..56..34..35..36..37..38..63..94.131.174.223.
.
..257.205.159.119..85..57..58..59..60..61..62..93.130.173.222.277
.
....258.206.160.120..86..87..88..89..90..91..92.129.172.221.276
.
......259.207.161.121.122.123.124.125.126.127.128.171.220.275
.
........260.208.162.163.164.165.166.167.168.169.170.219.274
.
..........261.209.210.211.212.213.214.215.216.217.218.273
.
............262.263.264.265.266.267.268.269.270.271.272
.
..............322.323.324.325.326.327.328.329.330.331
.
(End)
|
|
|
MAPLE
|
A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014
|
|
|
MATHEMATICA
|
LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* Harvey P. Dale, Nov 02 2011 *)
|
|
|
PROG
|
(PARI) a(n)=3*n^2-2*n+1 /* Michael Somos, Aug 03 2006 */
(MAGMA) [3*n^2 - 2*n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014
|
|
|
CROSSREFS
|
Cf. A054552 for example of square (or octagonal) spiral spoke.
Cf. A056106, A056107, A056108, A056109, A003215.
Sequence in context: A259702 A284923 A235707 * A212069 A106058 A086718
Adjacent sequences: A056102 A056103 A056104 * A056106 A056107 A056108
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
Henry Bottomley, Jun 09 2000
|
|
|
STATUS
|
approved
|
| |
|
|
