cp's OEIS Frontend

A056106 Second spoke of a hexagonal spiral.

%I A056106 #80 Sep 08 2022 08:45:01
%S A056106 1,3,11,25,45,71,103,141,185,235,291,353,421,495,575,661,753,851,955,
%T A056106 1065,1181,1303,1431,1565,1705,1851,2003,2161,2325,2495,2671,2853,
%U A056106 3041,3235,3435,3641,3853,4071,4295,4525,4761,5003,5251,5505,5765,6031,6303
%N A056106 Second spoke of a hexagonal spiral.
%C A056106 First differences of A027444. - _J. M. Bergot_, Jun 04 2012
%C A056106 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
%C A056106 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
%C A056106 For all n >= 6, a(n+1) expressed in base n is "353". - _Mathew Englander_, Jan 06 2021
%H A056106 Vincenzo Librandi, <a href="/A056106/b056106.txt">Table of n, a(n) for n = 0..10000</a>
%H A056106 Henry Bottomley, <a href="/A003215/a003215.gif">Spokes of a hexagonal spiral</a> (illustration of initial terms).
%H A056106 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.
%H A056106 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A056106 a(n) = 3*n^2 - n + 1.
%F A056106 a(n) = a(n-1) + 6*n - 4 = 2*a(n-1) - a(n-2) + 6.
%F A056106 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A056106 E.g.f.: (1+2*x+3*x^2)*exp(x). - _Paul Barry_, Mar 13 2003
%F A056106 a(n) = A096777(3*n) for n>0. - _Reinhard Zumkeller_, Dec 29 2007
%F A056106 G.f.: (1+5*x^2)/(1-3*x+3*x^2-x^3). - _Colin Barker_, Jan 04 2012
%F A056106 a(n) = n*A002061(n+1) - (n-1)*A002061(n). - _Bruno Berselli_, Jan 15 2013
%F A056106 a(-n) = A056108(n). - _Bruno Berselli_, Mar 13 2013
%t A056106 Table[3*n^2 - n + 1, {n,0,50}] (* _G. C. Greubel_, Jul 19 2017 *)
%o A056106 (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
%o A056106 (PARI) a(n) = 3*n^2-n+1;
%o A056106 (Haskell)
%o A056106 a056106 n = n * (3 * n - 1) + 1  -- _Reinhard Zumkeller_, Dec 13 2014
%Y A056106 First differences of A053698, A027444, and A188947.
%Y A056106 Cf. A113524 (semiprime terms), A002061.
%Y A056106 Cf. A001105, A001844, A058331, A251599.
%Y A056106 Other spokes: A003215, A056105, A056107, A056108, A056109.
%Y A056106 Other spirals: A054552.
%K A056106 nonn,easy
%O A056106 0,2
%A A056106 _Henry Bottomley_, Jun 09 2000