A056109 Fifth spoke of a hexagonal spiral.

1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321, 386, 457, 534, 617, 706, 801, 902, 1009, 1122, 1241, 1366, 1497, 1634, 1777, 1926, 2081, 2242, 2409, 2582, 2761, 2946, 3137, 3334, 3537, 3746, 3961, 4182, 4409, 4642, 4881, 5126, 5377, 5634, 5897, 6166, 6441

Offset: 0

Author

Henry Bottomley, Jun 09 2000

Keywords

• easy
• nonn

Comments

Squared distance from (0,0,-1) to (n,n,n) in R^3. - James R. Buddenhagen, Jun 15 2013

Examples

			Illustration of initial terms:
.
.                                                o
.                           o                 o o o o
.            o           o o o o           o o o o o o o
.   o     o o o o     o o o o o o o     o o o o o o o o o o
.            o           o o o o           o o o o o o o
.                           o                 o o o o
.                                                o
.
.   1        6              17                   34
- _Aaron David Fairbanks_, Feb 16 2025
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Henry Bottomley, Illustration of initial terms
• Tanya Khovanova, Recursive Sequences
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
• Leo Tavares, Triple Diamond Illustration
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A008810, A122430 (prime terms).

Other spokes: A003215, A056105, A056106, A056107, A056108.

Other spirals: A054552.

Cf. A000290.

Programs

GAP

List([0..50],n->3*n^2+2*n+1); # Muniru A Asiru, Oct 07 2018

Magma

[3*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Mar 15 2013

Maple

seq(coeff(series(n!*(exp(x)*(3*x^2+5*x+1)),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 07 2018

Mathematica

Table[3 n^2 + 2 n + 1, {n, 0, 100}] (* Vincenzo Librandi, Mar 15 2013 *)
LinearRecurrence[{3,-3,1},{1,6,17},60] (* Harvey P. Dale, Mar 28 2019 *)

PARI

{a(n) = 3*n^2 + 2*n + 1}; /* Michael Somos, Aug 03 2006 */

PARI

Vec((1+3*x+2*x^2)/(1-3*x+3*x^2-x^3)+O(x^100)) \\ Stefano Spezia, Oct 17 2018

Formula

a(n) = 3n^2+2n+1 = a(n-1)+6n-1 = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+4n = A056106(n)+3n = A056107(n)+2n = A056108(n)+n = A003215(n)-n.

G.f.: (1+3*x+2*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012

G.f.: (1 + x) * (1 + 2*x) / (1 - x)^3. - Michael Somos, Feb 04 2012

a(n) = A008810(3*n + 1) = A056105(-n). - Michael Somos, Aug 03 2006

E.g.f.: exp(x)*(1 + 5*x + 3*x^2). - Stefano Spezia, Oct 06 2018

a(n) = A000290(n+1) + 2*A000290(n). - Leo Tavares, May 29 2023

a(n) = A069894(n) - A000290(n+1). - Jarrod G. Sage, Jul 19 2024