# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a083884
Showing 1-1 of 1

%I A083884 #55 Sep 08 2022 08:45:10
%S A083884 1,5,41,365,3281,29525,265721,2391485,21523361,193710245,1743392201,
%T A083884 15690529805,141214768241,1270932914165,11438396227481,
%U A083884 102945566047325,926510094425921,8338590849833285,75047317648499561,675425858836496045,6078832729528464401
%N A083884 a(n) = (3^(2*n) + 1) / 2.
%C A083884 Number of compositions of even natural numbers into n parts <= 8. - _Adi Dani_, May 28 2011
%C A083884 a(n) for n >= 1 gives the number of line segments in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982) when parallel line segments that are connected end-to-end are counted as a single line segment. - _Jason V. Morgan_, Oct 08 2021
%D A083884 Siromoney, R., & Subramanian, K.G. (1982). Space-filling curves and infinite graphs. Graph-Grammars and Their Application to Computer Science.
%H A083884 Vincenzo Librandi, <a href="/A083884/b083884.txt">Table of n, a(n) for n = 0..200</a>
%H A083884 Roberto Amato, <a href="https://arxiv.org/abs/1912.05925">A note on Pythagorean Triples</a>, arXiv:1912.05925 [math.HO], 2019. See Example 2.1 p. 4.
%H A083884 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-9).
%F A083884 a(0) = 1, a(n) = 9*a(n-1) - 4.
%F A083884 a(n) = Sum_{k=0..n} binomial(2*n, 2*k)*4^k.
%F A083884 a(n) = A002438(n) / A000364(n); A000364(n) : Euler numbers.
%F A083884 G.f.: (1-5*x)/((1-x)*(1-9*x)).
%F A083884 a(n) = (3^n + 1^n + (-1)^n + (-3)^n)/4.
%F A083884 E.g.f.: exp(3*x) + exp(x) + exp(-x) + exp(-3*x).
%F A083884 Each term expresses a Pythagorean relationship, along with (a(n)-1) and a power of 3, n>0, such that sqrt((a(n))^2 - (a(n)-1)^2) = 3^n. E.g., 365^2 - 364^2 - 3^3 = 27 (the Pythagorean triangle (365, 364, 27)). - _Gary W. Adamson_, Jun 25 2006
%F A083884 a(n) = 10*a(n-1) - 9*a(n-2). - _Wesley Ivan Hurt_, Apr 21 2021
%e A083884 From _Adi Dani_, May 28 2011: (Start)
%e A083884 a(2)=41: there are 41 compositions of even natural numbers into 2 parts <=8:
%e A083884 (0,0);
%e A083884 (0,2),(2,0),(1,1);
%e A083884 (0,4),(4,0),(1,3),(3,1),(2,2);
%e A083884 (0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
%e A083884 (0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
%e A083884 (2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
%e A083884 (4,8),(8,4),(5,7),(7,5),(6,6);
%e A083884 (6,8),(8,6),(7,7);
%e A083884 (8,8).  (End)
%t A083884 f[n_] := (3^(2n)+1)/2; Table[f@i, {i,0,20}] (* _Michael De Vlieger_, Jan 28 2015 *)
%o A083884 (Magma) [(3^(2*n) + 1) / 2: n in [0..20]]; // _Vincenzo Librandi_, Jun 16 2011
%o A083884 (PARI) a(n)=(3^(2*n)+1)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A083884 Cf. A000364, A002438, A007853, A083885, A086645.
%K A083884 easy,nonn
%O A083884 0,2
%A A083884 _Paul Barry_, May 09 2003
%E A083884 Additional comments from _Philippe Deléham_, Jul 10 2005

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE