A139273 a(n) = n*(8*n - 3).

0, 5, 26, 63, 116, 185, 270, 371, 488, 621, 770, 935, 1116, 1313, 1526, 1755, 2000, 2261, 2538, 2831, 3140, 3465, 3806, 4163, 4536, 4925, 5330, 5751, 6188, 6641, 7110, 7595, 8096, 8613, 9146, 9695, 10260, 10841, 11438, 12051, 12680

Offset

0,2

Comments

Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139277 in the same spiral.

Also, sequence of numbers of the form d*A000217(n-1) + 5*n with generating functions x*(5+(d-5)*x)/(1-x)^3; the inverse binomial transform is 0,5,d,0,0,.. (0 continued). See Crossrefs. - Bruno Berselli, Feb 11 2011

Even decagonal numbers divided by 2. - Omar E. Pol, Aug 19 2011

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 8*n^2 - 3*n.

Sequences of the form a(n) = 8*n^2 + c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - R. J. Mathar, May 12 2008

a(n) = 16*n + a(n-1) - 11 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

From Bruno Berselli, Feb 11 2011: (Start)

G.f.: x*(5 + 11*x)/(1 - x)^3.

a(n) = 4*A000217(n) + A051866(n). (End)

a(n) = A028994(n)/2. - Omar E. Pol, Aug 19 2011

a(0)=0, a(1)=5, a(2)=26; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 02 2012

E.g.f.: (8*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 18 2017

Sum_{n>=1} 1/a(n) = 4*log(2)/3 - (sqrt(2)-1)*Pi/6 - sqrt(2)*arccoth(sqrt(2))/3. - Amiram Eldar, Jul 03 2020

Mathematica

Table[n (8 n - 3), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 26}, 40] (* Harvey P. Dale, Feb 02 2012 *)

Prog

(Magma) [ n*(8*n-3) : n in [0..40] ];  // Bruno Berselli, Feb 11 2011
(PARI) a(n)=n*(8*n-3) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734.

Sequence in context: A042883 A273701 A273709 * A185939 A273419 A273447

Adjacent sequences: A139270 A139271 A139272 * A139274 A139275 A139276

Keyword

nonn,easy

Author

Omar E. Pol, Apr 26 2008

Status

approved