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Number of compositions of 5*n-4 into parts 3 and 5.
5

%I #19 Mar 15 2024 07:24:59

%S 0,1,3,6,11,23,57,149,379,928,2227,5336,12872,31236,75949,184524,

%T 447702,1085401,2631240,6380241,15474230,37533077,91034937,220790480,

%U 535475968,1298668192,3149634952,7638811025,18526466357,44932341015,108974456212,264295580664

%N Number of compositions of 5*n-4 into parts 3 and 5.

%H Paolo Xausa, <a href="/A369848/b369848.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,-10,11,-5,1).

%F a(n) = A052920(5*n-4).

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

%F a(n) = 5*a(n-1) - 10*a(n-2) + 11*a(n-3) - 5*a(n-4) + a(n-5).

%F G.f.: x^2*(1-x)^2/((1-x)^5 - x^3).

%t LinearRecurrence[{5, -10, 11, -5, 1}, {0, 1, 3, 6, 11}, 50] (* _Paolo Xausa_, Mar 15 2024 *)

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

%Y Cf. A369804, A369845, A369846, A369847.

%Y Cf. A052920.

%K nonn,easy

%O 1,3

%A _Seiichi Manyama_, Feb 03 2024