A054556 a(n) = 4*n^2 - 9*n + 6.

1, 4, 15, 34, 61, 96, 139, 190, 249, 316, 391, 474, 565, 664, 771, 886, 1009, 1140, 1279, 1426, 1581, 1744, 1915, 2094, 2281, 2476, 2679, 2890, 3109, 3336, 3571, 3814, 4065, 4324, 4591, 4866, 5149, 5440, 5739, 6046, 6361, 6684, 7015, 7354, 7701, 8056, 8419, 8790

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

Move in 1-4 direction in a spiral organized like A068225 etc.

Equals binomial transform of [1, 3, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008

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

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

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• 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. A054555, A068225, A054552, A054554, A054567, A054569, A033951.

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.

Cf. A001105, A058331, A130883.

Programs

Magma

[4*n^2-9*n+6 : n in [1..50]]; // Vincenzo Librandi, Mar 10 2018

Maple

a:=n->4*n^2-9*n+6: seq(a(n),n=1..50); # Muniru A Asiru, Mar 09 2018

Mathematica

a[n_] := 4*n^2 - 9*n + 6; Array[a, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,4,15},50] (* Harvey P. Dale, Sep 06 2015 *)
CoefficientList[Series[-(6x^2 + x + 1)/(x - 1)^3, {x, 0, 49}], x] (* Robert G. Wilson v, Mar 12 2018 *)

PARI

a(n)=4*n^2-9*n+6 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n)^2 = Sum_{i = 0..2*(4*n-5)} (4*n^2-13*n+9+i)^2*(-1)^i = ((n-1)*(4*n-5)+1)^2. - Bruno Berselli, Apr 29 2010

From Harvey P. Dale, Aug 21 2011: (Start)

a(0)=1, a(1)=4, a(2)=15; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -x*(6*x^2+x+1)/(x-1)^3. (End)

From Franck Maminirina Ramaharo, Mar 09 2018: (Start)

a(n) = binomial(2*n - 2, 2) + 2*(n - 1)^2 + 1.

a(n) = A000384(n-1) + A058331(n-1).

a(n) = A130883(n-1) + A001105(n-1). (End)

E.g.f.: exp(x)*(6 - 5*x + 4*x^2) - 6. - Stefano Spezia, Apr 24 2024

Extensions

Edited by Frank Ellermann, Feb 24 2002

Incorrect formula deleted by N. J. A. Sloane, Aug 02 2009