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%I A054552 #80 Jun 01 2024 11:31:41
%S A054552 1,2,11,28,53,86,127,176,233,298,371,452,541,638,743,856,977,1106,
%T A054552 1243,1388,1541,1702,1871,2048,2233,2426,2627,2836,3053,3278,3511,
%U A054552 3752,4001,4258,4523,4796,5077,5366,5663,5968,6281,6602,6931,7268,7613,7966,8327
%N A054552 a(n) = 4*n^2 - 3*n + 1.
%C A054552 Also indices in any square spiral organized like A054551.
%C A054552 Equals binomial transform of [1, 1, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, May 11 2008
%C A054552 Ulam's spiral (E spoke). - _Robert G. Wilson v_, Oct 31 2011
%C A054552 For n > 0: left edge of the triangle A033293. - _Reinhard Zumkeller_, Jan 18 2012
%H A054552 Harvey P. Dale, <a href="/A054552/b054552.txt">Table of n, a(n) for n = 0..1000</a>
%H A054552 Scientific American, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue</a>
%H A054552 Leo Tavares, <a href="/A054552/a054552.jpg">Illustration: Hexagon/Square Pairs</a>
%H A054552 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A054552 G.f.: (1 - x + 8*x^2)/(1-x)^3.
%F A054552 a(n) = 8*n + a(n-1) - 7 (with a(0)=1). - _Vincenzo Librandi_, Aug 07 2010
%F A054552 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 A054552 E.g.f.: exp(x)*(1 + x + 4*x^2). - _Stefano Spezia_, May 14 2021
%F A054552 a(n) = A003215(n-1) + A000290(n). - _Leo Tavares_, Jul 21 2022
%e A054552 The spiral begins:
%e A054552 .
%e A054552 197-196-195-194-193-192-191-190-189-188-187-186-185-184-183
%e A054552   |                                                       |
%e A054552 198 145-144-143-142-141-140-139-138-137-136-135-134-133 182
%e A054552   |   |                                               |   |
%e A054552 199 146 101-100--99--98--97--96--95--94--93--92--91 132 181
%e A054552   |   |   |                                       |   |   |
%e A054552 200 147 102  65--64--63--62--61--60--59--58--57  90 131 180
%e A054552   |   |   |   |                               |   |   |   |
%e A054552 201 148 103  66  37--36--35--34--33--32--31  56  89 130 179
%e A054552   |   |   |   |   |                       |   |   |   |   |
%e A054552 202 149 104  67  38  17--16--15--14--13  30  55  88 129 178
%e A054552   |   |   |   |   |   |               |   |   |   |   |   |
%e A054552 203 150 105  68  39  18   5---4---3  12  29  54  87 128 177
%e A054552   |   |   |   |   |   |   |       |   |   |   |   |   |   |
%e A054552 204 151 106  69  40  19   6   1---2  11  28  53  86 127 176
%e A054552   |   |   |   |   |   |   |           |   |   |   |   |   |
%e A054552 205 152 107  70  41  20   7---8---9--10  27  52  85 126 175
%e A054552   |   |   |   |   |   |                   |   |   |   |   |
%e A054552 206 153 108  71  42  21--22--23--24--25--26  51  84 125 174
%e A054552   |   |   |   |   |                           |   |   |   |
%e A054552 207 154 109  72  43--44--45--46--47--48--49--50  83 124 173
%e A054552   |   |   |   |                                   |   |   |
%e A054552 208 155 110  73--74--75--76--77--78--79--80--81--82 123 172
%e A054552   |   |   |                                           |   |
%e A054552 209 156 111-112-113-114-115-116-117-118-119-120-121-122 171
%e A054552   |   |                                                   |
%e A054552 210 157-158-159-160-161-162-163-164-165-166-167-168-169-170
%e A054552   |
%e A054552 211-212-213-214-215-216-217-218-219-220-221-222-223-224-225
%e A054552 .
%e A054552 - _Robert G. Wilson v_, Jul 04 2014
%p A054552 A054552:=n->4*n^2-3*n+1: seq(A054552(n), n=0..50); # _Wesley Ivan Hurt_, Jul 11 2014
%t A054552 f[n_] := 4*n^2 - 3*n + 1; Array[f, 50, 0] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%t A054552 LinearRecurrence[{3,-3,1},{1,2,11},50] (* _Harvey P. Dale_, Jun 01 2024 *)
%o A054552 (PARI) a(n)= 4*n^2-3*n+1 \\ _Charles R Greathouse IV_, Jan 15 2012
%o A054552 (Magma) [4*n^2-3*n+1 : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 11 2014
%Y A054552 Cf. A033293, A054551, A108781.
%Y A054552 Spokes of square spiral: A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.
%Y A054552 Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y A054552 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 A054552 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 A054552 Cf. A003215.
%K A054552 easy,nonn
%O A054552 0,2
%A A054552 _Enoch Haga_ and _G. L. Honaker, Jr._, Apr 09 2000

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