A166147 a(n) = 4*n^2 + 4*n - 7.

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

Offset: 1

Author

Vincenzo Librandi, Oct 08 2009

Keywords

• nonn
• easy

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).
• Index entries for sequences related to cellular automata.
• Index to 2D 5-Neighbor Cellular Automata.
• Index to Elementary Cellular Automata.

Crossrefs

Cf. A028886.

Programs

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

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 *)

PARI

a(n)=8*binomial(n+1,2)-7 \\ Charles R Greathouse IV, Jan 11 2012

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

Extensions

New name from Charles R Greathouse IV, Jan 11 2012