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A056105 First spoke of a hexagonal spiral. 34
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 sequences related to 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

(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) - Robert G. Wilson v, Jul 05 2014

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

.

............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

.

- Robert G. Wilson v, Jul 05 2014

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: A237044 A248116 A235707 * A212069 A106058 A086718

Adjacent sequences:  A056102 A056103 A056104 * A056106 A056107 A056108

KEYWORD

easy,nonn

AUTHOR

Henry Bottomley, Jun 09 2000

STATUS

approved

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Last modified October 23 16:27 EDT 2014. Contains 248468 sequences.