OFFSET
0,2
COMMENTS
Sequence found by reading the line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Semi-axis opposite to A094159 in the same spiral.
Sum of the numbers from 2*n to 4*n. - Wesley Ivan Hurt, Nov 27 2015
From Peter M. Chema, Jan 21 2017: (Start)
Also 0 together with the partial sums of A017629.
Digit root is 0 together with period 3: repeat [9,3,9].
Final digits cycle a length period 10: repeat [0,9,0,3,8,5,4,5,8,3]. (End)
Sequence found by reading the line from 0, in the direction 0, 9, ..., in the triangle spiral. - Hans G. Oberlack, Dec 08 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Hans G. Oberlack, Triangle spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 6*n^2 + 3*n = 3*A014105(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, Oct 13 2013
G.f.: 3*x*(3+x) / (1-x)^3. - Wesley Ivan Hurt, Nov 27 2015
E.g.f.: 3*x*(2*x+3)*exp(x). - G. C. Greubel, Dec 07 2018
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(1 - log(2))/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/2 + log(2) - 2)/3. (End)
MAPLE
MATHEMATICA
Table[6n^2+3n, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 9, 30}, 50] (* Harvey P. Dale, Oct 13 2013 *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 27 2015 *)
PROG
(Magma) [3*n*(2*n+1): n in [0..50]]; // Vincenzo Librandi, Sep 20 2011
(PARI) a(n)=3*n*(2*n+1) \\ Charles R Greathouse IV, Oct 16 2015
(Sage) [3*n*(2*n+1) for n in range(50)] # G. C. Greubel, Dec 07 2018
(GAP) List([0..30], n -> 3*n*(2*n+1)); # G. C. Greubel, Dec 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Sep 17 2011
STATUS
approved