A054569 a(n) = 4*n^2 - 6*n + 3.

1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163, 2353, 2551, 2757, 2971, 3193, 3423, 3661, 3907, 4161, 4423, 4693, 4971, 5257, 5551, 5853, 6163, 6481, 6807, 7141, 7483, 7833, 8191

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

Move in 1-7 direction in a spiral organized like A068225 etc.

Third row of A082039. - Paul Barry, Apr 02 2003

Inverse binomial transform of A036826. - Paul Barry, Jun 11 2003

Equals the "middle sequence" T(2*n,n) of the Connell sequence A001614 as a triangle. - Johannes W. Meijer, May 20 2011

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

Links

• Harvey P. Dale, 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. A000384, A033951, A054552, A054554, A054556, A054567, A054568, A068225.

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

GAP

List([1..50], n-> 4*n^2-6*n+3); # G. C. Greubel, Jul 04 2019

Magma

[4*n^2-6*n+3: n in [1..50]]; // G. C. Greubel, Jul 04 2019

Mathematica

f[n_]:= 4*n^2-6*n+3; Array[f, 50] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
LinearRecurrence[{3,-3,1},{1,7,21},50] (* Harvey P. Dale, Nov 17 2012 *)

PARI

a(n)=4*n^2-6*n+3 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[4*n^2-6*n+3 for n in (1..50)] # G. C. Greubel, Jul 04 2019

Formula

a(n+1) = 4*n^2 + 2*n + 1. - Paul Barry, Apr 02 2003

a(n) = 4*n^2 - 6*n+3 - 3*0^n (with leading zero). - Paul Barry, Jun 11 2003

Binomial transform of [1, 6, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007

a(n) = 8*n + a(n-1) - 10 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010

From Colin Barker, Mar 23 2012: (Start)

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

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

a(n) = A000384(n) + A000384(n-1). - Bruce J. Nicholson, May 07 2017

E.g.f.: -3 + (3 - 2*x + 4*x^2)*exp(x). - G. C. Greubel, Jul 04 2019

Sum_{n>=1} 1/a(n) = A339237. - R. J. Mathar, Jan 22 2021

Extensions

Edited by Frank Ellermann, Feb 24 2002