A228888 a(n) = binomial(3*n + 2, 3).

10, 56, 165, 364, 680, 1140, 1771, 2600, 3654, 4960, 6545, 8436, 10660, 13244, 16215, 19600, 23426, 27720, 32509, 37820, 43680, 50116, 57155, 64824, 73150, 82160, 91881, 102340, 113564, 125580, 138415, 152096, 166650, 182104, 198485, 215820, 234136, 253460

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Formula

a(n) = binomial(3*n + 2, 3) = 1/6*(3*n)*(3*n + 1)*(3*n + 2).

a(-n) = - A006566(n).

a(n) = 1/6*A228889(n).

G.f.: (10*x + 16*x^2 + x^3)/(1 - x)^4 = 10*x + 56*x^2 + 165*x^3 + ....

Sum {n >= 1} 1/a(n) = 9/2 - 3/2*log(3) - 1/2*sqrt(3)*Pi.

Sum {n >= 1} (-1)^n/a(n) = 9/2 - 4*log(2) - 1/3*sqrt(3)*Pi.

Example

From Bruno Berselli, Jun 26 2018: (Start)
Including 0, row sums of the triangle:
| 0|   .................................................................. 0
| 1|   2   3   4   ..................................................... 10
| 5|   6   7   8   9  10  11   ......................................... 56
|12|  13  14  15  16  17  18  19  20  21   ............................ 165
|22|  23  24  25  26  27  28  29  30  31  32  33  34   ................ 364
|35|  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50   .... 680
...
in the first column of which we have the pentagonal numbers (A000326).
(End)

Maple

seq(binomial(3*n+2, 3), n = 1..38);

Mathematica

Table[(Binomial[3 n + 2, 3]), {n, 1, 40}] (* Vincenzo Librandi, Sep 09 2013 *)

Prog

(Magma) [Binomial(3*n + 2, 3): n in [1..40]]; // Vincenzo Librandi, Sep 09 2013

Crossrefs

Cf. A006566 (binomial(3*n,3)) and A228887 (binomial(3*n + 1,3)).

Cf. A228889.

Similar sequences are listed in A316224.

Sequence in context: A000814 A202071 A281207 * A137931 A053493 A198833

Adjacent sequences: A228885 A228886 A228887 * A228889 A228890 A228891

Keyword

nonn,easy

Author

Peter Bala, Sep 09 2013

Status

approved