A133694 a(n) = (3*n^2 + 3*n - 4)/2.

1, 7, 16, 28, 43, 61, 82, 106, 133, 163, 196, 232, 271, 313, 358, 406, 457, 511, 568, 628, 691, 757, 826, 898, 973, 1051, 1132, 1216, 1303, 1393, 1486, 1582, 1681, 1783, 1888, 1996, 2107, 2221, 2338, 2458, 2581, 2707, 2836, 2968, 3103, 3241, 3382, 3526, 3673

Offset: 1

Author

Gary W. Adamson, Sep 20 2007

Keywords

• nonn
• easy

Comments

Binomial transform of 1, 6, 3 followed by A000004, i.e., 1, 6, 3, 0, 0, 0, 0, ... .

Row sums of triangle A133981. - Gary W. Adamson, Sep 30 2007

Equals (1, 2, 3, 4, ...) convolved with (1, 5, 3, 3, 3, ...). Example: a(4) = (1, 2, 3, 4) dot (3, 3, 5, 1) = (3 + 6 + 15 + 4) = 28. - Gary W. Adamson, May 01 2009

Equivalently, numbers of the form 3*(h+1)*(2*h-1) + 1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 03 2017

Examples

			a(3) = 3*A000217(3) - 2 = 3*6 - 2 = 16.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Leo Tavares, Illustration: Triple Triangles.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000004, A000217, A133981.

Programs

Magma

a000217:=func; [3*a000217(n)-2: n in [1..60]];

Magma

[(3*n^2+3*n-4)/2: n in [1..50]]; // Vincenzo Librandi, Mar 30 2014

Maple

A133694:=n->(3*n^2 + 3*n - 4)/2; seq(A133694(n), n=1..30); # Wesley Ivan Hurt, Mar 26 2014

Mathematica

Table[(3*n^2 + 3*n - 4)/2, {n, 100}]
CoefficientList[Series[(1 + 4 x - 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 30 2014 *)
LinearRecurrence[{3,-3,1},{1,7,16},50] (* Harvey P. Dale, Sep 05 2020 *)

PARI

a(n)=(3*n^2+3*n-4)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 3*A000217(n) - 2.

a(n) = a(n-1) + 3*n for n > 1, a(1)=1. - Vincenzo Librandi, Nov 23 2010

G.f.: x*(1+4*x-2*x^2)/(1-x)^3. - Vincenzo Librandi, Mar 30 2014

Sum_{n>=1} 1/a(n) = 1/2 + 2*Pi*tan(sqrt(19/3)*Pi/2)/sqrt(57). - Amiram Eldar, Jun 08 2022

E.g.f.: 2 + exp(x)*(3*x*(2 + x) - 4)/2. - Stefano Spezia, Nov 23 2023

Extensions

Edited by Klaus Brockhaus, Nov 23 2010