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%I #52 Feb 22 2023 01:48:57
%S 0,20,60,120,200,300,420,560,720,900,1100,1320,1560,1820,2100,2400,
%T 2720,3060,3420,3800,4200,4620,5060,5520,6000,6500,7020,7560,8120,
%U 8700,9300,9920,10560,11220,11900,12600,13320,14060,14820,15600,16400,17220,18060,18920
%N a(n) = 10*n*(n+1).
%C 20 times the n-th triangular number.
%C 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
%C Numbers k such that 10*k + 25 is a square. - _Bruno Berselli_, May 14 2018
%H Vincenzo Librandi, <a href="/A163761/b163761.txt">Table of n, a(n) for n = 0..875</a>
%H John Elias, <a href="/A163761/a163761.png">Illustration of Initial Terms: Correlation of 10k+25 is a square</a>.
%H Shanzhen Gao and Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.
%H Shanzhen Gao and H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 20*A000217(n) = 10*A002378(n).
%F G.f.: 20*x/(1-x)^3.
%F E.g.f.: 10*x*(x+2)*exp(x). - _G. C. Greubel_, Aug 03 2017
%F From _Amiram Eldar_, Feb 22 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 1/10.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/10.
%F Product_{n>=1} (1 - 1/a(n)) = -(10/Pi)*cos(sqrt(7/5)*Pi/2).
%F Product_{n>=1} (1 + 1/a(n)) = (10/Pi)*cos(sqrt(3/5)*Pi/2). (End)
%t LinearRecurrence[{3,-3,1},{0,20,60}, 50] (* or *) Table[10*n*(n+1), {n,0,50}] (* _G. C. Greubel_, Aug 03 2017 *)
%o (Magma) [10*n*(n+1): n in [0..50]];
%o (PARI) a(n)=10*n*(n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A002378.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Aug 03 2009
%E Entries checked by _R. J. Mathar_, Aug 06 2009