A014255 Expansion of (1+2*x+3*x^2)/((1-x)*(1-x^2)^2).

1, 3, 8, 12, 21, 27, 40, 48, 65, 75, 96, 108, 133, 147, 176, 192, 225, 243, 280, 300, 341, 363, 408, 432, 481, 507, 560, 588, 645, 675, 736, 768, 833, 867, 936, 972, 1045, 1083, 1160, 1200, 1281, 1323, 1408, 1452, 1541, 1587, 1680, 1728, 1825, 1875, 1976, 2028

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A002620(n+1) is the n-th partial arithmetic mean. - Michael Somos, Feb 14 2004

The smallest integer greater than a(n-1) such that the n-th partial arithmetic mean is an integer is a(n) if n is odd or a(n)-(n+1) if n is even. - Michael Somos, Feb 14 2004

Beginning with 1, the smallest integer greater than the previous term such that no three consecutive terms are in arithmetic progression and the n-th partial arithmetic mean is an integer. - Amarnath Murthy, Feb 05 2004

The maximum possible number of black cells in a solution to an (n+1) X (n+1) nurikabe grid. - Tanya Khovanova, Feb 24 2009

Let M = an infinite lower triangular matrix with alternate columns composed of (1,1,1,...) and (1,2,2,2,...); and Q = the diagonalized variant of (1,2,3,...). Then Q*M = a triangle with row sums = A014255. - Gary W. Adamson, May 14 2010

Number of pairs (x,y) with x and y in {0,...,n} having the same parity and x+y < n. - Clark Kimberling, Jul 02 2012

Form an array with m(0,0)=0 and m(i,j)=|i^2 - j^2|. One-half the difference between the sum of the terms in antidiagonal(n) and those in antidiagonal(n-1)=a(n). - J. M. Bergot, Jul 10 2013

For n > 0, a(n-1) is the sum of the largest parts in the partitions of 2n into two odd parts. - Wesley Ivan Hurt, Dec 19 2017

Sum of the odd numbers in the interval [m, 2*m] with m > 0. Example: for m = 5, the sum of the odd numbers in [5, 10] is 5 + 7 + 9 = 21, therefore 21 is a term of this sequence. - Bruno Berselli, Oct 25 2018

Examples

			From _Gary W. Adamson_, May 14 2010: (Start)
The first few rows of the generating triangle are
  1;
  1,  2;
  1,  4,  3;
  1,  4,  3,  4;
  1,  4,  3,  8,  5;
  1,  4,  3,  8,  5,  6;
  1,  4,  3,  8,  5, 12,  7;
  1,  4,  3,  8,  5, 12,  7,  8;
  1,  4,  3,  8,  5, 12,  7, 16,  9;
  1,  4,  3,  8,  5, 12,  7, 16,  9, 10;
  ...
Row sums  are 1, 3, 8, 12, 21, 27, 40, ... (End)
G.f. = 1 + 3*x + 8*x^2 + 12*x^3 + 21*x^4 + 27*x^5 + 40*x^6 + 48*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Programs

GAP

List([0..55], n-> (6*n^2+14*n+7 +(-1)^n*(2*n+1))/8); # G. C. Greubel, Jun 18 2019

Magma

[(6*n^2+14*n+7 +(-1)^n*(2*n+1))/8: n in [0..55]]; // G. C. Greubel, Jun 18 2019

Mathematica

Array[(# + 1)^2 - Floor[(# + 1)/2]^2 &, 52, 0] (* or *)
CoefficientList[Series[(1+2x+3x^2)/((1-x)(1-x^2)^2), {x, 0, 51}], x] (* Michael De Vlieger, Dec 20 2017 *)

PARI

vector(55, n, n--; (6*n^2+14*n+7 +(-1)^n*(2*n+1))/8) \\ G. C. Greubel, Jun 18 2019

Sage

[(6*n^2+14*n+7 +(-1)^n*(2*n+1))/8 for n in (0..55)] # G. C. Greubel, Jun 18 2019

Formula

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

a(n) = (n+1)^2 - floor((n+1)/2)^2. - Franklin T. Adams-Watters, May 26 2006

a(n) = (6*n^2 + 14*n + 7 + (-1)^n*(2*n + 1))/8. - R. J. Mathar, Mar 22 2011

a(n) = (k+1)*(3*k+1) if n = 2*k, 3*(k+1)^2 if n = 2*k+1. - Michael Somos, Feb 27 2014

E.g.f.: ((4+9*x+3*x^2)*cosh(x) + (3+11*x+3*x^2)*sinh(x))/4. - G. C. Greubel, Jun 18 2019