A056106 Second spoke of a hexagonal spiral.

1, 3, 11, 25, 45, 71, 103, 141, 185, 235, 291, 353, 421, 495, 575, 661, 753, 851, 955, 1065, 1181, 1303, 1431, 1565, 1705, 1851, 2003, 2161, 2325, 2495, 2671, 2853, 3041, 3235, 3435, 3641, 3853, 4071, 4295, 4525, 4761, 5003, 5251, 5505, 5765, 6031, 6303

Offset: 0

Author

Henry Bottomley, Jun 09 2000

Keywords

• nonn
• easy

Comments

First differences of A027444. - J. M. Bergot, Jun 04 2012

Numbers of the form ((h^2+h+1)^2+(-h^2+h+1)^2+(h^2+h-1)^2)/(h^2-h+1) for h=n-1. - Bruno Berselli, Mar 13 2013

For n > 0: 2*a(n) = A058331(n) + A001105(n) + A001844(n-1) = A251599(3*n-2) + A251599(3*n-1) + A251599(3*n). - Reinhard Zumkeller, Dec 13 2014

For all n >= 6, a(n+1) expressed in base n is "353". - Mathew Englander, Jan 06 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Henry Bottomley, Spokes of a hexagonal spiral (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).

Crossrefs

First differences of A053698, A027444, and A188947.

Cf. A113524 (semiprime terms), A002061.

Cf. A001105, A001844, A058331, A251599.

Other spokes: A003215, A056105, A056107, A056108, A056109.

Other spirals: A054552.

Programs

Haskell

a056106 n = n * (3 * n - 1) + 1  -- Reinhard Zumkeller, Dec 13 2014

Magma

I:=[1,3]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+6: n in [1..50]]; // Vincenzo Librandi, Nov 14 2011

Mathematica

Table[3*n^2 - n + 1, {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

PARI

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

Formula

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

a(n) = a(n-1) + 6*n - 4 = 2*a(n-1) - a(n-2) + 6.

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

E.g.f.: (1+2*x+3*x^2)*exp(x). - Paul Barry, Mar 13 2003

a(n) = A096777(3*n) for n>0. - Reinhard Zumkeller, Dec 29 2007

G.f.: (1+5*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012

a(n) = n*A002061(n+1) - (n-1)*A002061(n). - Bruno Berselli, Jan 15 2013

a(-n) = A056108(n). - Bruno Berselli, Mar 13 2013