A054552 a(n) = 4*n^2 - 3*n + 1.

1, 2, 11, 28, 53, 86, 127, 176, 233, 298, 371, 452, 541, 638, 743, 856, 977, 1106, 1243, 1388, 1541, 1702, 1871, 2048, 2233, 2426, 2627, 2836, 3053, 3278, 3511, 3752, 4001, 4258, 4523, 4796, 5077, 5366, 5663, 5968, 6281, 6602, 6931, 7268, 7613, 7966, 8327

Offset: 0

Author

Enoch Haga and G. L. Honaker, Jr., Apr 09 2000

Keywords

• easy
• nonn

Comments

Also indices in any square spiral organized like A054551.

Equals binomial transform of [1, 1, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 11 2008

Ulam's spiral (E spoke). - Robert G. Wilson v, Oct 31 2011

For n > 0: left edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012

Examples

			The spiral begins:
.
197-196-195-194-193-192-191-190-189-188-187-186-185-184-183
  |                                                       |
198 145-144-143-142-141-140-139-138-137-136-135-134-133 182
  |   |                                               |   |
199 146 101-100--99--98--97--96--95--94--93--92--91 132 181
  |   |   |                                       |   |   |
200 147 102  65--64--63--62--61--60--59--58--57  90 131 180
  |   |   |   |                               |   |   |   |
201 148 103  66  37--36--35--34--33--32--31  56  89 130 179
  |   |   |   |   |                       |   |   |   |   |
202 149 104  67  38  17--16--15--14--13  30  55  88 129 178
  |   |   |   |   |   |               |   |   |   |   |   |
203 150 105  68  39  18   5---4---3  12  29  54  87 128 177
  |   |   |   |   |   |   |       |   |   |   |   |   |   |
204 151 106  69  40  19   6   1---2  11  28  53  86 127 176
  |   |   |   |   |   |   |           |   |   |   |   |   |
205 152 107  70  41  20   7---8---9--10  27  52  85 126 175
  |   |   |   |   |   |                   |   |   |   |   |
206 153 108  71  42  21--22--23--24--25--26  51  84 125 174
  |   |   |   |   |                           |   |   |   |
207 154 109  72  43--44--45--46--47--48--49--50  83 124 173
  |   |   |   |                                   |   |   |
208 155 110  73--74--75--76--77--78--79--80--81--82 123 172
  |   |   |                                           |   |
209 156 111-112-113-114-115-116-117-118-119-120-121-122 171
  |   |                                                   |
210 157-158-159-160-161-162-163-164-165-166-167-168-169-170
  |
211-212-213-214-215-216-217-218-219-220-221-222-223-224-225
.
- _Robert G. Wilson v_, Jul 04 2014
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Scientific American, Cover of the March 1964 issue
• Leo Tavares, Illustration: Hexagon/Square Pairs
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A033293, A054551, A108781.

Spokes of square spiral: A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.

Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

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.

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.

Cf. A003215.

Programs

Magma

[4*n^2-3*n+1 : n in [0..50]]; // Wesley Ivan Hurt, Jul 11 2014

Maple

A054552:=n->4*n^2-3*n+1: seq(A054552(n), n=0..50); # Wesley Ivan Hurt, Jul 11 2014

Mathematica

f[n_] := 4*n^2 - 3*n + 1; Array[f, 50, 0] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,2,11},50] (* Harvey P. Dale, Jun 01 2024 *)

PARI

a(n)= 4*n^2-3*n+1 \\ Charles R Greathouse IV, Jan 15 2012

Formula

G.f.: (1 - x + 8*x^2)/(1-x)^3.

a(n) = 8*n + a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 07 2010

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

E.g.f.: exp(x)*(1 + x + 4*x^2). - Stefano Spezia, May 14 2021

a(n) = A003215(n-1) + A000290(n). - Leo Tavares, Jul 21 2022