A054554 a(n) = 4*n^2 - 10*n + 7.

1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071, 2257, 2451, 2653, 2863, 3081, 3307, 3541, 3783, 4033, 4291, 4557, 4831, 5113, 5403, 5701, 6007, 6321, 6643, 6973, 7311, 7657, 8011, 8373, 8743

Offset: 1

Author

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

Keywords

• easy
• nonn

Comments

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

Equals binomial transform of [1, 2, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008

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

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

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• James Grime and Brady Haran, Prime Spirals, Numberphile video (2013).
• Scientific American, Cover of the March 1964 issue
• Leo Tavares, Illustration: Hexagonal Dual Rays
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Cf. A014105.

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, A227776.

Programs

Maple

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

Mathematica

f[n_] := 4n^2 -10n + 7; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,3,13},60] (* Harvey P. Dale, Jan 20 2025 *)

PARI

a(n)=4*n^2-10*n+7 \\ Charles R Greathouse IV, Nov 05 2013

Formula

a(n) = 8*n + a(n-1) - 14 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 07 2010

G.f.: -x*(7*x^2+1)/(x-1)^3. - Colin Barker, Sep 21 2012

For n > 2, a(n) = A014105(n) + A014105(n-1). - Bruce J. Nicholson, May 07 2017

From Leo Tavares, Feb 21 2022: (Start)

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

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

a(k+1) = 4k^2 - 2k + 1 in the Numberphile video. - Frank Ellermann, Mar 11 2020

E.g.f.: exp(x)*(7 - 6*x + 4*x^2) - 7. - Stefano Spezia, Apr 24 2024

Extensions

Edited by Frank Ellermann, Feb 24 2002