%I #77 Apr 24 2024 19:51:27
%S 1,4,15,34,61,96,139,190,249,316,391,474,565,664,771,886,1009,1140,
%T 1279,1426,1581,1744,1915,2094,2281,2476,2679,2890,3109,3336,3571,
%U 3814,4065,4324,4591,4866,5149,5440,5739,6046,6361,6684,7015,7354,7701,8056,8419,8790
%N a(n) = 4*n^2 - 9*n + 6.
%C Move in 1-4 direction in a spiral organized like A068225 etc.
%C Equals binomial transform of [1, 3, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 30 2008
%C Ulam's spiral (N spoke). - _Robert G. Wilson v_, Oct 31 2011
%C Also, numbers of the form m*(4*m+1)+1 for nonpositive m. - _Bruno Berselli_, Jan 06 2016
%H Ivan Panchenko, <a href="/A054556/b054556.txt">Table of n, a(n) for n = 1..1000</a>
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%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)^2 = Sum_{i = 0..2*(4*n-5)} (4*n^2-13*n+9+i)^2*(-1)^i = ((n-1)*(4*n-5)+1)^2. - _Bruno Berselli_, Apr 29 2010
%F From _Harvey P. Dale_, Aug 21 2011: (Start)
%F a(0)=1, a(1)=4, a(2)=15; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F G.f.: -x*(6*x^2+x+1)/(x-1)^3. (End)
%F From _Franck Maminirina Ramaharo_, Mar 09 2018: (Start)
%F a(n) = binomial(2*n - 2, 2) + 2*(n - 1)^2 + 1.
%F a(n) = A000384(n-1) + A058331(n-1).
%F a(n) = A130883(n-1) + A001105(n-1). (End)
%F E.g.f.: exp(x)*(6 - 5*x + 4*x^2) - 6. - _Stefano Spezia_, Apr 24 2024
%p a:=n->4*n^2-9*n+6: seq(a(n),n=1..50); # _Muniru A Asiru_, Mar 09 2018
%t a[n_] := 4*n^2 - 9*n + 6; Array[a, 40] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%t LinearRecurrence[{3,-3,1},{1,4,15},50] (* _Harvey P. Dale_, Sep 06 2015 *)
%t CoefficientList[Series[-(6x^2 + x + 1)/(x - 1)^3, {x, 0, 49}], x] (* _Robert G. Wilson v_, Mar 12 2018 *)
%o (PARI) a(n)=4*n^2-9*n+6 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [4*n^2-9*n+6 : n in [1..50]]; // _Vincenzo Librandi_, Mar 10 2018
%Y Cf. A054555, A068225, A054552, A054554, A054567, A054569, A033951.
%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.
%Y Cf. A001105, A058331, A130883.
%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 Incorrect formula deleted by _N. J. A. Sloane_, Aug 02 2009