A139271 a(n) = 2*n*(4*n-3).

0, 2, 20, 54, 104, 170, 252, 350, 464, 594, 740, 902, 1080, 1274, 1484, 1710, 1952, 2210, 2484, 2774, 3080, 3402, 3740, 4094, 4464, 4850, 5252, 5670, 6104, 6554, 7020, 7502, 8000, 8514, 9044, 9590, 10152, 10730, 11324, 11934, 12560, 13202, 13860, 14534, 15224

Offset: 0

Author

Omar E. Pol, Apr 26 2008

Keywords

• easy
• nonn
• walk

Comments

Sequence found by reading the line from 0, in the direction 0, 2, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A033585 in the same spiral.

Twice decagonal numbers (or twice 10-gonal numbers). - Omar E. Pol, May 15 2008

a(n) is the number of walks in a cubic lattice of n dimensions that reach the point of origin for the first time after 4 steps. - Shel Kaphan, Mar 20 2023

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).

Crossrefs

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Cf. A001107.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=16). - Bruno Berselli, Jun 10 2013

Row n=2 of A361397.

Programs

Mathematica

Table[8n^2-6n,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,2,20},50] (* Harvey P. Dale, Sep 26 2016 *)

PARI

a(n)=2*n*(4*n-3) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 8*n^2 - 6*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) = 3a(n-1) - 3a(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) = A001107(n)*2. - Omar E. Pol, May 15 2008

a(n) = 16*n + a(n-1) - 14 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010

From G. C. Greubel, Jul 18 2017: (Start)

G.f.: (2*x)*(7*x+1)/(1-x)^3.

E.g.f.: (8*x^2 + 2*x)*exp(x). (End)

Sum_{n>=1} 1/a(n) = Pi/12 + log(2)/2. - Amiram Eldar, Mar 28 2023

Extensions

Corrected by Harvey P. Dale, Sep 26 2016