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Number of compositions of 5*n-2 into parts 3 and 5.
5

%I #16 Mar 15 2024 11:35:02

%S 1,2,3,5,12,34,92,230,549,1299,3109,7536,18364,44713,108575,263178,

%T 637699,1545839,3749001,9093989,22058847,53501860,129755543,314685488,

%U 763192224,1850966760,4489176073,10887655332,26405874658,64042115197,155321124452

%N Number of compositions of 5*n-2 into parts 3 and 5.

%H Paolo Xausa, <a href="/A369847/b369847.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-2).

%F a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,n-1-3*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*(1-x)^3/((1-x)^5 - x^3).

%t LinearRecurrence[{5, -10, 11, -5, 1}, {1, 2, 3, 5, 12}, 50] (* _Paolo Xausa_, Mar 15 2024 *)

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

%Y Cf. A369804, A369845, A369846, A369848.

%Y Cf. A052920.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Feb 03 2024