|
%I #24 Nov 12 2022 02:11:37
%S 0,3,24,171,1200,8403,58824,411771,2882400,20176803,141237624,
%T 988663371,6920643600,48444505203,339111536424,2373780754971,
%U 16616465284800,116315256993603,814206798955224,5699447592686571,39896133148806000
%N a(n) = (7^n - 1)/2.
%C Number of compositions of odd natural numbers into n parts < 7. - _Adi Dani_, Jun 11 2011
%H Vincenzo Librandi, <a href="/A120741/b120741.txt">Table of n, a(n) for n = 0..300</a>
%H Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-7).
%F a(n) = A034494(n) - 1.
%F a(n) = 8*a(n-1) - 7*a(n-2), n >= 2.
%F a(n) = right term in M^n * [1,0], where M is the 2 X 2 matrix [4,3; 3,4].
%F From _G. C. Greubel_, Nov 11 2022: (Start)
%F G.f.: 3*x/((1-x)*(1-7*x)).
%F E.g.f.: (1/2)*(exp(7*x) - exp(x)). (End)
%e From _Adi Dani_, Jun 11 2011: (Start)
%e a(2)=24: there are 24 compositions of odd numbers into 2 parts < 7:
%e 1: (0,1), (1,0);
%e 3: (0,3), (3,0), (1,2), (2,1);
%e 5: (0,5), (5,0), (1,4), (4,1), (2,3), (3,2);
%e 7: (1,6), (6,1), (2,5), (5,2), (3,4), (4,3);
%e 9: (3,6), (6,3), (4,5), (5,4);
%e 11: (5,6),(6,5). (End)
%e a(4) = 1200 = A034494(4) - 1, where A034494(4) = 1201.
%e a(4) = 1200 = 8*a(3) - 7*a(2) = 8*171 - 7*24.
%e a(4) = 1200 = right term in M^n * [1,0] = [A034494(4), a(4)] = [1201, 1200].
%t Table[1/2*(7^n - 1), {n, 0, 25}]
%o (Magma) [(7^n-1)/2: n in [0..25]]; // _Vincenzo Librandi_, Jun 11 2011
%o (PARI) a(n)=7^n\2 \\ _Charles R Greathouse IV_, Jun 11 2011
%o (SageMath) [(7^n-1)/2 for n in range(31)] # _G. C. Greubel_, Nov 11 2022
%Y Cf. A034494.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Jun 30 2006
%E Complete edit by _Joerg Arndt_, Jun 11 2011
|
