This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A054554 #77 Jul 29 2025 20:03:43
%S A054554 1,3,13,31,57,91,133,183,241,307,381,463,553,651,757,871,993,1123,
%T A054554 1261,1407,1561,1723,1893,2071,2257,2451,2653,2863,3081,3307,3541,
%U A054554 3783,4033,4291,4557,4831,5113,5403,5701,6007,6321,6643,6973,7311,7657,8011,8373,8743
%N A054554 a(n) = 4*n^2 - 10*n + 7.
%C A054554 Move in 1-3 direction in a spiral organized like A068225 etc.
%C A054554 Equals binomial transform of [1, 2, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, May 03 2008
%C A054554 Ulam's spiral (NE spoke). - _Robert G. Wilson v_, Oct 31 2011
%C A054554 Number of ternary strings of length 2*(n-1) that have one or no 0's, one or no 1's, and an even number of 2's. For n=2, the 3 strings of length 2 are 01, 10 and 22. For n=3, the 13 strings of length 4 are the 12 permutations of 0122 and 2222. - _Enrique Navarrete_, Jul 25 2025
%H A054554 Ivan Panchenko, <a href="/A054554/b054554.txt">Table of n, a(n) for n = 1..1000</a>
%H A054554 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=iFuR97YcSLM">Prime Spirals</a>, Numberphile video (2013).
%H A054554 Scientific American, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue</a>
%H A054554 Leo Tavares, <a href="/A054554/a054554.jpg">Illustration: Hexagonal Dual Rays</a>
%H A054554 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A054554 a(n) = 8*n + a(n-1) - 14 with n > 1, a(1)=1. - _Vincenzo Librandi_, Aug 07 2010
%F A054554 G.f.: -x*(7*x^2+1)/(x-1)^3. - _Colin Barker_, Sep 21 2012
%F A054554 For n > 2, a(n) = A014105(n) + A014105(n-1). - _Bruce J. Nicholson_, May 07 2017
%F A054554 From _Leo Tavares_, Feb 21 2022: (Start)
%F A054554 a(n) = A003215(n-2) + 2*A000217(n-1). See Hexagonal Dual Rays illustration in links.
%F A054554 a(n) = A227776(n-1) - 4*A000217(n-1). (End)
%F A054554 a(k+1) = 4k^2 - 2k + 1 in the Numberphile video. - _Frank Ellermann_, Mar 11 2020
%F A054554 E.g.f.: exp(x)*(7 - 6*x + 4*x^2) - 7. - _Stefano Spezia_, Apr 24 2024
%p A054554 A054554:=n->4*n^2 -10*n + 7; seq(A054554(k),k=1..100); # _Wesley Ivan Hurt_, Nov 05 2013
%t A054554 f[n_] := 4n^2 -10n + 7; Array[f, 40] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%t A054554 LinearRecurrence[{3,-3,1},{1,3,13},60] (* _Harvey P. Dale_, Jan 20 2025 *)
%o A054554 (PARI) a(n)=4*n^2-10*n+7 \\ _Charles R Greathouse IV_, Nov 05 2013
%Y A054554 Cf. A054553, A068225, A054552, A054556, A054567, A054569, A033951.
%Y A054554 Cf. A014105.
%Y A054554 Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y A054554 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 A054554 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 A054554 Cf. A003215, A227776.
%K A054554 easy,nonn
%O A054554 1,2
%A A054554 _Enoch Haga_, _G. L. Honaker, Jr._, Apr 10 2000
%E A054554 Edited by _Frank Ellermann_, Feb 24 2002