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a(n) = 4*n^2 - 7*n + 4.
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%I #65 Aug 03 2024 19:23:04

%S 1,6,19,40,69,106,151,204,265,334,411,496,589,690,799,916,1041,1174,

%T 1315,1464,1621,1786,1959,2140,2329,2526,2731,2944,3165,3394,3631,

%U 3876,4129,4390,4659,4936,5221,5514,5815,6124,6441,6766,7099,7440,7789,8146,8511,8884

%N a(n) = 4*n^2 - 7*n + 4.

%C The number 1 is placed in the middle of a sheet of squared paper and the numbers 2, 3, 4, 5, 6, etc. are written in a clockwise spiral around 1, as in A068225 etc. This sequence is read off along one of the rays from 1.

%C Ulam's spiral (W spoke of A054552). - _Robert G. Wilson v_, Oct 31 2011

%C Also, numbers of the form m*(4*m+1)+1 for nonnegative m. - _Bruno Berselli_, Jan 06 2016

%C The sequence forms the 1x2 diagonal of the square maze arrangement in A081344. - _Jarrod G. Sage_, Jul 17 2024

%H Ivan Panchenko, <a href="/A054567/b054567.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) = 8*n+a(n-1)-11 for n>1, a(1)=1. - _Vincenzo Librandi_, Aug 07 2010

%F a(n) = A204674(n-1) / n. - _Reinhard Zumkeller_, Jan 18 2012

%F From _Colin Barker_, Oct 25 2014: (Start)

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

%F G.f.: -x*(4*x^2+3*x+1) / (x-1)^3. (End)

%F E.g.f.: exp(x)*(4 - 3*x + 4*x^2) - 4. - _Stefano Spezia_, Apr 24 2024

%F a(n) = A016742(n-1) + n. - _Jarrod G. Sage_, Jul 17 2024

%t Table[4 n^2 - 7 n + 4, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)

%o (PARI) Vec(-x*(4*x^2+3*x+1)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Oct 25 2014

%Y Cf. A054566, A068225, A054552, A054554, A054556, A054569, A033951, A204674.

%Y Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...

%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 nonn,easy

%O 1,2

%A _Enoch Haga_, _G. L. Honaker, Jr._, Apr 10 2000

%E Edited by _Frank Ellermann_, Feb 24 2002

%E Typo fixed by _Charles R Greathouse IV_, Oct 28 2009