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

0,2

Comments

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

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).

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

Maple

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

Mathematica

s=1; lst={s}; Do[s+=n+5; AppendTo[lst, s], {n, 0, 6!, 6}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 04 2008 *)
Table[3 n^2 + 2 n + 1, {n, 0, 100}] (* Vincenzo Librandi, Mar 15 2013 *)
CoefficientList[Series[E^x (1 + 5 x + 3 x^2) , {x, 0, 20}], x]*Table[k!, {k, 0, 100}] (* Stefano Spezia, Oct 06 2018 *)
LinearRecurrence[{3, -3, 1}, {1, 6, 17}, 60] (* Harvey P. Dale, Mar 28 2019 *)

Prog

(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
(Magma) [3*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Mar 15 2013
(GAP) List([0..50], n->3*n^2+2*n+1); # Muniru A Asiru, Oct 07 2018
(Python) for n in range(0, 100): print(int(3*n**2 + 2*n + 1), end=', ') # Stefano Spezia, Oct 16 2018

Crossrefs

Cf. A008810, A122430 (prime terms).

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

Other spirals: A054552.

Cf. A000290.

Sequence in context: A301711 A066486 A301719 * A023545 A038633 A083045

Adjacent sequences: A056106 A056107 A056108 * A056110 A056111 A056112

Keyword

easy,nonn

Author

Henry Bottomley, Jun 09 2000

Status

approved