OFFSET
1,2
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
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).
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
EXAMPLE
a(3) = 3*A000217(3) - 2 = 3*6 - 2 = 16.
MAPLE
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 *)
PROG
(Magma) a000217:=func<n | n*(n+1) div 2>; [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
(PARI) a(n)=(3*n^2+3*n-4)/2 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Sep 20 2007
EXTENSIONS
Edited by Klaus Brockhaus, Nov 23 2010
STATUS
approved