A163761 a(n) = 10*n*(n+1).

0, 20, 60, 120, 200, 300, 420, 560, 720, 900, 1100, 1320, 1560, 1820, 2100, 2400, 2720, 3060, 3420, 3800, 4200, 4620, 5060, 5520, 6000, 6500, 7020, 7560, 8120, 8700, 9300, 9920, 10560, 11220, 11900, 12600, 13320, 14060, 14820, 15600, 16400, 17220, 18060, 18920

Offset: 0

Author

Vincenzo Librandi, Aug 03 2009

Keywords

• nonn
• easy

Comments

20 times the n-th triangular number.

a(n) is the number of one-sided n-step prudent walks, from (0,0) to (3,3), for n-6 is even. - Shanzhen Gao, Apr 26 2011

Numbers k such that 10*k + 25 is a square. - Bruno Berselli, May 14 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..875
• John Elias, Illustration of Initial Terms: Correlation of 10k+25 is a square.
• Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
• Shanzhen Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A002378.

Programs

Magma

[10*n*(n+1): n in [0..50]];

Mathematica

LinearRecurrence[{3,-3,1},{0,20,60}, 50] (* or *) Table[10*n*(n+1), {n,0,50}] (* G. C. Greubel, Aug 03 2017 *)

PARI

a(n)=10*n*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 20*A000217(n) = 10*A002378(n).

G.f.: 20*x/(1-x)^3.

E.g.f.: 10*x*(x+2)*exp(x). - G. C. Greubel, Aug 03 2017

From Amiram Eldar, Feb 22 2023: (Start)

Sum_{n>=1} 1/a(n) = 1/10.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/10.

Product_{n>=1} (1 - 1/a(n)) = -(10/Pi)*cos(sqrt(7/5)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (10/Pi)*cos(sqrt(3/5)*Pi/2). (End)

Extensions

Entries checked by R. J. Mathar, Aug 06 2009