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

Author

Omar E. Pol, Sep 17 2011

Keywords

• nonn
• easy

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).

Crossrefs

Bisection of A045943.

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

Programs

GAP

List([0..30], n -> 3*n*(2*n+1)); # G. C. Greubel, Dec 07 2018

Magma

[3*n*(2*n+1): n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

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 *)

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

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)