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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

COMMENTS

Also the number of (not necessarily maximum) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

H. Bottomley, Illustration of initial terms

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Eric Weisstein's World of Mathematics, Clique

Eric Weisstein's World of Mathematics, Grid Graph

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 3*n^2 - 2*n + 1.

a(n) = a(n-1) + 6*n - 5.

a(n) = 2*a(n-1) - a(n-2) + 6.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

a(n) = A056106(n) - n = A056107(n) - 2*n.

a(n) = A056108(n) - 3*n = A056109(n) - 4*n = A003215(n) - 5*n.

A008810(3*n-1) = A056109(-n) = a(n). - Michael Somos, Aug 03 2006.

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

a(n) = A000290(n-1) + 2*A000290(n), n >= 1. - J. M. Bergot, Mar 03 2018

E.g.f.: (1 + x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018

EXAMPLE

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 *)

Table[3 n^2 - 2 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)

CoefficientList[Series[(-1 + x - 6 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

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

(Sage) [3*n^2-2*n+1 for n in range(50)] # G. C. Greubel, Dec 02 2018

(GAP) List([0..50], n -> 3*n^2-2*n+1); # G. C. Greubel, Dec 02 2018

CROSSREFS

Cf. A054552 for example of square (or octagonal) spiral spoke.

Cf. A056106, A056107, A056108, A056109, A003215.

Sequence in context: A259702 A284923 A235707 * A323891 A212069 A106058

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 December 11 18:46 EST 2019. Contains 329925 sequences. (Running on oeis4.)