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%I #33 Jun 29 2023 16:01:17
%S 1,4,25,172,1201,8404,58825,411772,2882401,20176804,141237625,
%T 988663372,6920643601,48444505204,339111536425,2373780754972,
%U 16616465284801,116315256993604,814206798955225,5699447592686572
%N a(n) = (7^n+1)/2.
%C Binomial transform of A081341. Inverse binomial transform of A081342. [_R. J. Mathar_, Oct 23 2008]
%C Number of compositions of even natural numbers into n parts <=6. [_Adi Dani_, May 28 2011]
%C From _Charlie Marion_, Jun 24 2011: (Start)
%C a(n)+(a(n)+1)+...+(a(n+1)-7^n-1)=(a(n+1)-7^n)+...+(a(n+1)-1). Let
%C S(2n) and S(2n+1) be the sets of addends on the left- and right-hand
%C sides, respectively, of the preceding equations. Then, since the
%C intersection of any 2 different S(i) is null and the union of all of
%C them is the positive integers, {S(i)} forms a partition of the
%C positive integers. See also A034659.
%C In general, for k>0, let b(n)=((4k+3)^n+1)/2. Then b(n)+(b(n)+1)+ ...
%C +(b(n+1)-(4k+3)^n-1)=k*((b(n+1)-(4k+3)^n)+ ... +(b(n+1)-1)). Then,
%C for each k, the set of addends on the two sides of these equations
%C also forms a partition of the positive integers. Also, with b(0)=1,
%C b(n)=(4k+3)*b(n-1)-(2k+1).
%C For k>0, let c(0)=1 and, for n>0, c(n)=(2*(2k+1))^n/2. Then the
%C sequence b(0),b(1),... is the binomial transform of the sequence
%C c(0),c(1),....
%C For k>0, let d(2n)=(2k+1)^(2n) and d(2n+1)=0. Then the sequence
%C b(0),b(1),... is the (2k+2)nd binomial transform of the sequence
%C d(0),d(1),.... (End)
%H Vincenzo Librandi, <a href="/A034494/b034494.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8, -7).
%F E.g.f.: exp(4*x)*cosh(3*x). - _Paul Barry_, Apr 20 2003
%F a(n) = 7a(n-1) - 3, a(0) = 1.
%F G.f.: (1-4*x)/((1-x)*(1-7*x)). - _Philippe Deléham_, Jul 11 2005
%F a(n) = 8*a(n-1)-7*a(n-2), a(0)=1, a(1)=4. [_Philippe Deléham_, Nov 15 2008]
%F a(n) = ((4+sqrt(9))^n+(4-sqrt(9))^n)/2. [Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008]
%e From _Adi Dani_, May 28 2011: (Start)
%e a(2)=25: there are 25 compositions of even numbers into 2 parts <=6:
%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 (2,6),(6,2),(3,5),(5,3),(4,4)
%e (4,6),(6,4),(5,5)
%e (6,6)
%e (end)
%p A034494:=n->(7^n+1)/2: seq(A034494(n), n=0..30); # _Wesley Ivan Hurt_, Apr 09 2017
%o (Magma) [(7^n+1)/2: n in [0..30]]; // _Vincenzo Librandi_, Jun 16 2011
%o (PARI) a(n)=(7^n+1)/2 \\ _Charles R Greathouse IV_, Jul 02 2013
%o (PARI) Vec((1-4*x)/((1-x)*(1-7*x)) + O(x^100)) \\ _Altug Alkan_, Nov 01 2015
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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