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

0,2

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).

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

Mathematica

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

Prog

(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
(PARI) a(n) = 3*n^2-n+1;
(Haskell)
a056106 n = n * (3 * n - 1) + 1  -- Reinhard Zumkeller, Dec 13 2014

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.

Sequence in context: A185258 A118436 A293413 * A320035 A147382 A212971

Adjacent sequences: A056103 A056104 A056105 * A056107 A056108 A056109

Keyword

nonn,easy

Author

Henry Bottomley, Jun 09 2000

Status

approved