A195319 - OEIS

%I #66 Mar 14 2023 10:16:01

%S 0,9,30,63,108,165,234,315,408,513,630,759,900,1053,1218,1395,1584,

%T 1785,1998,2223,2460,2709,2970,3243,3528,3825,4134,4455,4788,5133,

%U 5490,5859,6240,6633,7038,7455,7884,8325,8778,9243,9720,10209,10710,11223

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

%C 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.

%C Sum of the numbers from 2*n to 4*n. - _Wesley Ivan Hurt_, Nov 27 2015

%C From _Peter M. Chema_, Jan 21 2017: (Start)

%C Also 0 together with the partial sums of A017629.

%C Digit root is 0 together with period 3: repeat [9,3,9].

%C Final digits cycle a length period 10: repeat [0,9,0,3,8,5,4,5,8,3]. (End)

%C Sequence found by reading the line from 0, in the direction 0, 9, ..., in the triangle spiral. - _Hans G. Oberlack_, Dec 08 2018

%H Vincenzo Librandi, <a href="/A195319/b195319.txt">Table of n, a(n) for n = 0..10000</a>

%H Hans G. Oberlack, <a href="/A195319/a195319.png">Triangle spiral</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 6*n^2 + 3*n = 3*A014105(n).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Harvey P. Dale_, Oct 13 2013

%F G.f.: 3*x*(3+x) / (1-x)^3. - _Wesley Ivan Hurt_, Nov 27 2015

%F a(n) = A000217(3*n) + 3*A000217(n). - _Bruno Berselli_, Aug 31 2017

%F E.g.f.: 3*x*(2*x+3)*exp(x). - _G. C. Greubel_, Dec 07 2018

%F From _Amiram Eldar_, Feb 27 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 2*(1 - log(2))/3.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/2 + log(2) - 2)/3. (End)

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

%t Table[6n^2+3n,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,9,30},50] (* _Harvey P. Dale_, Oct 13 2013 *)

%t CoefficientList[Series[3 x (3 + x)/(1 - x)^3, {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Nov 27 2015 *)

%o (Magma) [3*n*(2*n+1): n in [0..50]]; // _Vincenzo Librandi_, Sep 20 2011

%o (PARI) a(n)=3*n*(2*n+1) \\ _Charles R Greathouse IV_, Oct 16 2015

%o (Sage) [3*n*(2*n+1) for n in range(50)] # _G. C. Greubel_, Dec 07 2018

%o (GAP) List([0..30], n -> 3*n*(2*n+1)); # _G. C. Greubel_, Dec 07 2018

%Y Bisection of A045943.

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

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Sep 17 2011