OFFSET
1,2
COMMENTS
Primes in the sequence are in A028886. - Bruno Berselli, Mar 16 2012
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
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Stephen Wolfram, A New Kind of Science.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = a(n-1)+8*n with n>1, a(1)=1.
From Vincenzo Librandi, Mar 15 2012: (Start)
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 *)
Table[4 n^2 + 4 n - 7, {n, 46}] (* Michael De Vlieger, Apr 27 2016 *)
PROG
(PARI) a(n)=8*binomial(n+1, 2)-7 \\ Charles R Greathouse IV, Jan 11 2012
(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
Vincenzo Librandi, Oct 08 2009
EXTENSIONS
New name from Charles R Greathouse IV, Jan 11 2012
STATUS
approved