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Number of compositions of 5*n-1 into parts 2 and 5.
5

%I #16 Mar 15 2024 11:09:30

%S 1,3,7,18,52,154,450,1301,3753,10838,31327,90568,261813,756786,

%T 2187496,6323023,18277014,52830706,152709940,441415867,1275934888,

%U 3688154521,10660798289,30815580241,89074003241,257472939209,744238632362,2151259638423,6218325456983

%N Number of compositions of 5*n-1 into parts 2 and 5.

%H Paolo Xausa, <a href="/A369842/b369842.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,10,-5,1).

%F a(n) = A001687(5*n).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n+1+3*k,n-1-2*k).

%F a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

%F G.f.: x*(1-x)^2/((1-x)^5 - x^2).

%t LinearRecurrence[{5, -9, 10, -5, 1}, {1, 3, 7, 18, 52}, 50] (* _Paolo Xausa_, Mar 15 2024 *)

%o (PARI) a(n) = sum(k=0, n\2, binomial(n+1+3*k, n-1-2*k));

%Y Cf. A369803, A369840, A369843, A369844.

%Y Cf. A001687.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Feb 03 2024