|
%I #68 Sep 08 2022 08:45:01
%S 1,7,21,43,73,111,157,211,273,343,421,507,601,703,813,931,1057,1191,
%T 1333,1483,1641,1807,1981,2163,2353,2551,2757,2971,3193,3423,3661,
%U 3907,4161,4423,4693,4971,5257,5551,5853,6163,6481,6807,7141,7483,7833,8191
%N a(n) = 4*n^2 - 6*n + 3.
%C Move in 1-7 direction in a spiral organized like A068225 etc.
%C Third row of A082039. - _Paul Barry_, Apr 02 2003
%C Inverse binomial transform of A036826. - _Paul Barry_, Jun 11 2003
%C Equals the "middle sequence" T(2*n,n) of the Connell sequence A001614 as a triangle. - _Johannes W. Meijer_, May 20 2011
%C Ulam's spiral (SW spoke). - _Robert G. Wilson v_, Oct 31 2011
%H Harvey P. Dale, <a href="/A054569/b054569.txt">Table of n, a(n) for n = 1..1000</a>
%H Robert G. Wilson v, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue of Scientific American</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n+1) = 4*n^2 + 2*n + 1. - _Paul Barry_, Apr 02 2003
%F a(n) = 4*n^2 - 6*n+3 - 3*0^n (with leading zero). - _Paul Barry_, Jun 11 2003
%F Binomial transform of [1, 6, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 28 2007
%F a(n) = 8*n + a(n-1) - 10 (with a(1)=1). - _Vincenzo Librandi_, Aug 07 2010
%F From _Colin Barker_, Mar 23 2012: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F G.f.: x*(1+x)*(1+3*x)/(1-x)^3. (End)
%F a(n) = A000384(n) + A000384(n-1). - _Bruce J. Nicholson_, May 07 2017
%F E.g.f.: -3 + (3 - 2*x + 4*x^2)*exp(x). - _G. C. Greubel_, Jul 04 2019
%F Sum_{n>=1} 1/a(n) = A339237. - _R. J. Mathar_, Jan 22 2021
%t f[n_]:= 4*n^2-6*n+3; Array[f, 50] (* _Vladimir Joseph Stephan Orlovsky_, Sep 02 2008 *)
%t LinearRecurrence[{3,-3,1},{1,7,21},50] (* _Harvey P. Dale_, Nov 17 2012 *)
%o (PARI) a(n)=4*n^2-6*n+3 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [4*n^2-6*n+3: n in [1..50]]; // _G. C. Greubel_, Jul 04 2019
%o (Sage) [4*n^2-6*n+3 for n in (1..50)] # _G. C. Greubel_, Jul 04 2019
%o (GAP) List([1..50], n-> 4*n^2-6*n+3) # _G. C. Greubel_, Jul 04 2019
%Y Cf. A000384, A033951, A054552, A054554, A054556, A054567, A054568, A068225.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%K easy,nonn
%O 1,2
%A _Enoch Haga_, _G. L. Honaker, Jr._, Apr 10 2000
%E Edited by _Frank Ellermann_, Feb 24 2002
|
