A195319 Three times second hexagonal numbers: 3*n*(2*n+1).

0, 9, 30, 63, 108, 165, 234, 315, 408, 513, 630, 759, 900, 1053, 1218, 1395, 1584, 1785, 1998, 2223, 2460, 2709, 2970, 3243, 3528, 3825, 4134, 4455, 4788, 5133, 5490, 5859, 6240, 6633, 7038, 7455, 7884, 8325, 8778, 9243, 9720, 10209, 10710, 11223

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

a(n) = A000217(3*n) + 3*A000217(n). - Bruno Berselli, Aug 31 2017

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

A195319:=n->6*n^2 + 3*n: seq(A195319(n), n=0..50); # Wesley Ivan Hurt, Nov 27 2015

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

Bisection of A045943.

Cf. A000217, A001318, A014105, A094159, A017629.

Sequence in context: A063150 A063161 A295867 * A073399 A225275 A005919

Adjacent sequences: A195316 A195317 A195318 * A195320 A195321 A195322

Keyword

nonn,easy

Author

Omar E. Pol, Sep 17 2011

Status

approved