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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
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.
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
Cf. A028886.
Sequence in context: A163185 A138005 A267421 * A028886 A146443 A110226
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Oct 08 2009
EXTENSIONS
New name from Charles R Greathouse IV, Jan 11 2012
STATUS
approved