A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900

Offset: 0

Author

Creighton Dement, Feb 17 2005

Keywords

• nonn
• easy

Comments

A floretion-generated sequence.

a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006

Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010

Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011

a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020

One odd followed by three evens.

From Paul Curtz, Mar 06 2020: (Start)

b(n) = 0, 1, 6, 16, 30, 49, ... = 0, a(n).

( 25, 12, 4, 0, 1, 6, 16, 30, ...

-13, -8, -4 1, 5, 10, 14, 19, ...

5, 4, 5, 4, 5, 4, 5, 4, ... .)

b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .

A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.

Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

Programs

Magma

[(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

Mathematica

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)

PARI

a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

Formula

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

a(2n) = A016778(n) = (3n+1)^2.

a(n) + a(n+1) = A038764(n+1).

a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006

a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011

a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011

From Paul Curtz, Mar 04 2020: (Start)

a(n) = A006578(n) + A001859(n) + A077043(n+1).

a(n) = A274221(2+2*n).

a(20+n) - a(n) = 30*(32+3*n).

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

a(n) = A047237(n) * A047251(n).

a(n) = A001651(n+1) * A032766(n).(End)

E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2011