A163756 14 times triangular numbers.

0, 14, 42, 84, 140, 210, 294, 392, 504, 630, 770, 924, 1092, 1274, 1470, 1680, 1904, 2142, 2394, 2660, 2940, 3234, 3542, 3864, 4200, 4550, 4914, 5292, 5684, 6090, 6510, 6944, 7392, 7854, 8330, 8820, 9324, 9842, 10374, 10920, 11480, 12054, 12642, 13244, 13860

Offset

0,2

Comments

Sequence found by reading the line from 0, in the direction 0, 14, ... and the same line from 0, in the direction 0, 42, ..., in the square spiral whose vertices are the generalized 16-gonal numbers. - Omar E. Pol, Oct 03 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = 7*n*(n+1) = 14*A000217(n).

G.f.: 14*x/(1-x)^3.

a(n) = 7*A002378(n) = 2*A024966(n) = A069127(n+1) - 1. - Omar E. Pol, Oct 03 2011

E.g.f.: 7*x*(x + 2)*exp(x). - G. C. Greubel, Aug 02 2017

From Amiram Eldar, Feb 21 2023: (Start)

Sum_{n>=1} 1/a(n) = 1/7.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/7.

Product_{n>=1} (1 - 1/a(n)) = -(7/Pi)*cos(sqrt(11/7)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (7/Pi)*cos(sqrt(3/7)*Pi/2). (End)

Mathematica

Table[7*n*(n-1), {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
14*Accumulate[Range[0, 50]] (* or *) LinearRecurrence[{3, -3, 1}, {0, 14, 42}, 50] (* Harvey P. Dale, May 11 2021 *)

Prog

(PARI) a(n)=7*n*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A000217, A002378, A069127.

Cf. A274978 (generalized 16-gonal numbers).

Sequence in context: A208359 A245629 A356452 * A005587 A244101 A212514

Adjacent sequences: A163753 A163754 A163755 * A163757 A163758 A163759

Keyword

nonn,easy

Author

Vincenzo Librandi, Aug 03 2009

Extensions

Extended by R. J. Mathar, Aug 06 2009

Status

approved