|
|
A166147
|
|
a(n) = 4*n^2 + 4*n - 7.
|
|
6
|
|
|
1, 17, 41, 73, 113, 161, 217, 281, 353, 433, 521, 617, 721, 833, 953, 1081, 1217, 1361, 1513, 1673, 1841, 2017, 2201, 2393, 2593, 2801, 3017, 3241, 3473, 3713, 3961, 4217, 4481, 4753, 5033, 5321, 5617, 5921, 6233, 6553, 6881, 7217, 7561, 7913, 8273, 8641
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016
a(n) = y - x for any primitive Pythagorean triangle (x^2 + y^2 = z^2), where z - x = 8. Also, a(n+2) = y + x, and y = 8n + 12. - Boyd Blundell, Jul 31 2021
|
|
REFERENCES
|
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = a(n-1)+8*n with n>1, a(1)=1.
G.f.: x*(1+14*x-7*x^2)/(1-x)^3.
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). (End)
E.g.f.: (-7 + 8*x + 4*x^2)*exp(x) + 7. - G. C. Greubel, Apr 26 2016
Sum_{n>=1} 1/a(n) = 1/7 + (Pi/(8*sqrt(2)))*tan(sqrt(2)*Pi). - Amiram Eldar, Feb 20 2023
|
|
MATHEMATICA
|
CoefficientList[Series[(1+14x-7x^2)/(1-x)^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3, -3, 1}, {1, 17, 41}, 50] (* Vincenzo Librandi, Mar 15 2012 *)
|
|
PROG
|
(Magma) I:=[1, 17, 41]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 15 2012
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|