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Number of compositions of n into parts 4 and 5 with at least one 4 and one 5.
2

%I #12 Dec 21 2023 11:46:23

%S 0,0,0,0,0,0,0,0,0,2,0,0,0,3,3,0,0,4,6,4,0,5,10,10,5,6,15,20,15,13,21,

%T 35,35,29,35,56,70,65,64,92,126,136,129,156,219,263,265,285,375,484,

%U 528,550,660,860,1013,1078,1210,1521,1873,2092,2288,2732,3394

%N Number of compositions of n into parts 4 and 5 with at least one 4 and one 5.

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1, -1, 1, 3, 3, 3, 2, -1, -2, -2, -2, -1).

%F a(n) = a(n-4)+a(n-5)+b(n) where b(n) is the 20-cycle (1,0,0,1,0,1,0,1,0,0,1,1,0,0,0,2,0,0,0,1) and b(n)=b(n-20). Initial values are b(14)=1, a(9)=2, a(10)=0, a(11)=0, a(12)=0, a(13)=3.

%F G.f.: 1+1/(1-x^5-x^4)-1/(1-x^5)-1/(1-x^4) (see comment A245332). - courtesy of _Joerg Arndt_

%e a(22)=10 The tuples are (55444)(54544)(54454)... where a(22)=5!/3!2!=10.

%t CoefficientList[Series[1 + 1/(1 - x^5 - x^4) - 1/(1 - x^5) - 1/(1 - x^4), {x, 0, 60}], x] (* _Wesley Ivan Hurt_, Jul 26 2014 *)

%Y Cf. A245332, A245492, A245487.

%K nonn,easy

%O 0,10

%A _David Neil McGrath_, Jul 25 2014