%I #55 Sep 08 2022 08:45:10

%S 1,5,41,365,3281,29525,265721,2391485,21523361,193710245,1743392201,

%T 15690529805,141214768241,1270932914165,11438396227481,

%U 102945566047325,926510094425921,8338590849833285,75047317648499561,675425858836496045,6078832729528464401

%N a(n) = (3^(2*n) + 1) / 2.

%C Number of compositions of even natural numbers into n parts <= 8. - _Adi Dani_, May 28 2011

%C 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 Siromoney, R., & Subramanian, K.G. (1982). Space-filling curves and infinite graphs. Graph-Grammars and Their Application to Computer Science.

%H Vincenzo Librandi, <a href="/A083884/b083884.txt">Table of n, a(n) for n = 0..200</a>

%H 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-9).

%F a(0) = 1, a(n) = 9*a(n-1) - 4.

%F a(n) = Sum_{k=0..n} binomial(2*n, 2*k)*4^k.

%F a(n) = A002438(n) / A000364(n); A000364(n) : Euler numbers.

%F G.f.: (1-5*x)/((1-x)*(1-9*x)).

%F a(n) = (3^n + 1^n + (-1)^n + (-3)^n)/4.

%F E.g.f.: exp(3*x) + exp(x) + exp(-x) + exp(-3*x).

%F 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 a(n) = 10*a(n-1) - 9*a(n-2). - _Wesley Ivan Hurt_, Apr 21 2021

%e From _Adi Dani_, May 28 2011: (Start)

%e a(2)=41: there are 41 compositions of even natural numbers into 2 parts <=8:

%e (0,0);

%e (0,2),(2,0),(1,1);

%e (0,4),(4,0),(1,3),(3,1),(2,2);

%e (0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);

%e (0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);

%e (2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);

%e (4,8),(8,4),(5,7),(7,5),(6,6);

%e (6,8),(8,6),(7,7);

%e (8,8). (End)

%t f[n_] := (3^(2n)+1)/2; Table[f@i, {i,0,20}] (* _Michael De Vlieger_, Jan 28 2015 *)

%o (Magma) [(3^(2*n) + 1) / 2: n in [0..20]]; // _Vincenzo Librandi_, Jun 16 2011

%o (PARI) a(n)=(3^(2*n)+1)/2 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A000364, A002438, A007853, A083885, A086645.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 09 2003

%E Additional comments from _Philippe Deléham_, Jul 10 2005