A054567 a(n) = 4*n^2 - 7*n + 4.

1, 6, 19, 40, 69, 106, 151, 204, 265, 334, 411, 496, 589, 690, 799, 916, 1041, 1174, 1315, 1464, 1621, 1786, 1959, 2140, 2329, 2526, 2731, 2944, 3165, 3394, 3631, 3876, 4129, 4390, 4659, 4936, 5221, 5514, 5815, 6124, 6441, 6766, 7099, 7440, 7789, 8146, 8511, 8884

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

The number 1 is placed in the middle of a sheet of squared paper and the numbers 2, 3, 4, 5, 6, etc. are written in a clockwise spiral around 1, as in A068225 etc. This sequence is read off along one of the rays from 1.

Ulam's spiral (W spoke of A054552). - Robert G. Wilson v, Oct 31 2011

Also, numbers of the form m*(4*m+1)+1 for nonnegative m. - Bruno Berselli, Jan 06 2016

The sequence forms the 1x2 diagonal of the square maze arrangement in A081344. - Jarrod G. Sage, Jul 17 2024

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A054566, A068225, A054552, A054554, A054556, A054569, A033951, A204674.

Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...

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.

Programs

Mathematica

Table[4 n^2 - 7 n + 4, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

PARI

Vec(-x*(4*x^2+3*x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Oct 25 2014

Formula

a(n) = 8*n+a(n-1)-11 for n>1, a(1)=1. - Vincenzo Librandi, Aug 07 2010

a(n) = A204674(n-1) / n. - Reinhard Zumkeller, Jan 18 2012

From Colin Barker, Oct 25 2014: (Start)

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

G.f.: -x*(4*x^2+3*x+1) / (x-1)^3. (End)

E.g.f.: exp(x)*(4 - 3*x + 4*x^2) - 4. - Stefano Spezia, Apr 24 2024

a(n) = A016742(n-1) + n. - Jarrod G. Sage, Jul 17 2024

Extensions

Edited by Frank Ellermann, Feb 24 2002

Typo fixed by Charles R Greathouse IV, Oct 28 2009