%I #71 Jan 20 2025 11:02:31

%S 1,3,13,31,57,91,133,183,241,307,381,463,553,651,757,871,993,1123,

%T 1261,1407,1561,1723,1893,2071,2257,2451,2653,2863,3081,3307,3541,

%U 3783,4033,4291,4557,4831,5113,5403,5701,6007,6321,6643,6973,7311,7657,8011,8373,8743

%N a(n) = 4*n^2 - 10*n + 7.

%C Move in 1-3 direction in a spiral organized like A068225 etc.

%C Equals binomial transform of [1, 2, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, May 03 2008

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

%H Ivan Panchenko, <a href="/A054554/b054554.txt">Table of n, a(n) for n = 1..1000</a>

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=iFuR97YcSLM">Prime Spirals</a>, Numberphile video (2013).

%H Scientific American, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue</a>

%H Leo Tavares, <a href="/A054554/a054554.jpg">Illustration: Hexagonal Dual Rays</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) - 14 with n > 1, a(1)=1. - _Vincenzo Librandi_, Aug 07 2010

%F G.f.: -x*(7*x^2+1)/(x-1)^3. - _Colin Barker_, Sep 21 2012

%F For n > 2, a(n) = A014105(n) + A014105(n-1). - _Bruce J. Nicholson_, May 07 2017

%F From _Leo Tavares_, Feb 21 2022: (Start)

%F a(n) = A003215(n-2) + 2*A000217(n-1). See Hexagonal Dual Rays illustration in links.

%F a(n) = A227776(n-1) - 4*A000217(n-1). (End)

%F a(k+1) = 4k^2 - 2k + 1 in the Numberphile video. - _Frank Ellermann_, Mar 11 2020

%F E.g.f.: exp(x)*(7 - 6*x + 4*x^2) - 7. - _Stefano Spezia_, Apr 24 2024

%p A054554:=n->4*n^2 -10*n + 7; seq(A054554(k),k=1..100); # _Wesley Ivan Hurt_, Nov 05 2013

%t f[n_] := 4n^2 -10n + 7; Array[f, 40] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)

%t LinearRecurrence[{3,-3,1},{1,3,13},60] (* _Harvey P. Dale_, Jan 20 2025 *)

%o (PARI) a(n)=4*n^2-10*n+7 \\ _Charles R Greathouse IV_, Nov 05 2013

%Y Cf. A054553, A068225, A054552, A054556, A054567, A054569, A033951.

%Y Cf. A014105.

%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. A003215, A227776.

%K easy,nonn

%O 1,2

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

%E Edited by _Frank Ellermann_, Feb 24 2002