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

Offset

1,2

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).

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

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). - Harvey P. Dale, Aug 21 2011

G.f.: -(6*x^2+x+1)/(x-1)^3. - Harvey P. Dale, Aug 21 2011

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)

Maple

a:=n->4*n^2-9*n+6: seq(a(n), n=0..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 *)

Prog

(PARI) a(n)=4*n^2-9*n+6 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [4*n^2-9*n+6 : n in [1..50]]; // Vincenzo Librandi, Mar 10 2018

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.

Sequence in context: A022265 A120389 A124150 * A113693 A211537 A213420

Adjacent sequences: A054553 A054554 A054555 * A054557 A054558 A054559

Keyword

nonn,easy

Author

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

Extensions

Edited by Frank Ellermann, Feb 24 2002

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

Status

approved