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%I #80 Jun 01 2024 11:31:41
%S 1,2,11,28,53,86,127,176,233,298,371,452,541,638,743,856,977,1106,
%T 1243,1388,1541,1702,1871,2048,2233,2426,2627,2836,3053,3278,3511,
%U 3752,4001,4258,4523,4796,5077,5366,5663,5968,6281,6602,6931,7268,7613,7966,8327
%N a(n) = 4*n^2 - 3*n + 1.
%C Also indices in any square spiral organized like A054551.
%C Equals binomial transform of [1, 1, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, May 11 2008
%C Ulam's spiral (E spoke). - _Robert G. Wilson v_, Oct 31 2011
%C For n > 0: left edge of the triangle A033293. - _Reinhard Zumkeller_, Jan 18 2012
%H Harvey P. Dale, <a href="/A054552/b054552.txt">Table of n, a(n) for n = 0..1000</a>
%H Scientific American, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue</a>
%H Leo Tavares, <a href="/A054552/a054552.jpg">Illustration: Hexagon/Square Pairs</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1 - x + 8*x^2)/(1-x)^3.
%F a(n) = 8*n + a(n-1) - 7 (with a(0)=1). - _Vincenzo Librandi_, Aug 07 2010
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=11. - _Harvey P. Dale_, Oct 10 2011
%F E.g.f.: exp(x)*(1 + x + 4*x^2). - _Stefano Spezia_, May 14 2021
%F a(n) = A003215(n-1) + A000290(n). - _Leo Tavares_, Jul 21 2022
%e The spiral begins:
%e .
%e 197-196-195-194-193-192-191-190-189-188-187-186-185-184-183
%e | |
%e 198 145-144-143-142-141-140-139-138-137-136-135-134-133 182
%e | | | |
%e 199 146 101-100--99--98--97--96--95--94--93--92--91 132 181
%e | | | | | |
%e 200 147 102 65--64--63--62--61--60--59--58--57 90 131 180
%e | | | | | | | |
%e 201 148 103 66 37--36--35--34--33--32--31 56 89 130 179
%e | | | | | | | | | |
%e 202 149 104 67 38 17--16--15--14--13 30 55 88 129 178
%e | | | | | | | | | | | |
%e 203 150 105 68 39 18 5---4---3 12 29 54 87 128 177
%e | | | | | | | | | | | | | |
%e 204 151 106 69 40 19 6 1---2 11 28 53 86 127 176
%e | | | | | | | | | | | | |
%e 205 152 107 70 41 20 7---8---9--10 27 52 85 126 175
%e | | | | | | | | | | |
%e 206 153 108 71 42 21--22--23--24--25--26 51 84 125 174
%e | | | | | | | | |
%e 207 154 109 72 43--44--45--46--47--48--49--50 83 124 173
%e | | | | | | |
%e 208 155 110 73--74--75--76--77--78--79--80--81--82 123 172
%e | | | | |
%e 209 156 111-112-113-114-115-116-117-118-119-120-121-122 171
%e | | |
%e 210 157-158-159-160-161-162-163-164-165-166-167-168-169-170
%e |
%e 211-212-213-214-215-216-217-218-219-220-221-222-223-224-225
%e .
%e - _Robert G. Wilson v_, Jul 04 2014
%p A054552:=n->4*n^2-3*n+1: seq(A054552(n), n=0..50); # _Wesley Ivan Hurt_, Jul 11 2014
%t f[n_] := 4*n^2 - 3*n + 1; Array[f, 50, 0] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%t LinearRecurrence[{3,-3,1},{1,2,11},50] (* _Harvey P. Dale_, Jun 01 2024 *)
%o (PARI) a(n)= 4*n^2-3*n+1 \\ _Charles R Greathouse IV_, Jan 15 2012
%o (Magma) [4*n^2-3*n+1 : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 11 2014
%Y Cf. A033293, A054551, A108781.
%Y Spokes of square spiral: A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.
%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.
%K easy,nonn
%O 0,2
%A _Enoch Haga_ and _G. L. Honaker, Jr._, Apr 09 2000