%I #55 Sep 08 2022 08:45:56

%S 1,1,3,8,21,54,140,363,941,2439,6322,16387,42476,110100,285385,739733,

%T 1917427,4970072,12882689,33392610,86555408,224356187,581543081,

%U 1507390367,3907235410,10127760455,26251689768,68045765768,176378217169,457181650329,1185038973363

%N a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), a(-2)=0, a(-1)=0, a(0)=1, a(1)=1.

%C Number of compositions of n where there is one sort of part 1, two sorts of part 2, three sorts of part 3, and four sorts of every other part. - _Joerg Arndt_, Mar 15 2014

%H G. C. Greubel, <a href="/A190139/b190139.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,1,1)

%F G.f.: (1-x)/(1-2*x-x^2-x^3-x^4).

%F a(n) = Sum_{k=1..n} (Sum_{t=k..n} (Sum_{j=0..k} C(k,j) * Sum_{i=j..t-k+j} C(j,i-j)*C(k-j,t-3*(k-j)-i)*C(-t+n+k-1,k-1))), n>0, a(0)=1.

%F a(n) = A103142(n) - A103142(n-1). - _R. J. Mathar_, Sep 17 2011

%t RecurrenceTable[{a[n] == 2 a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4], a[-2] == a[-1] == 0, a[0] == a[1] == 1}, a, {n, 0, 30}] (* _Michael De Vlieger_, Oct 28 2015 *)

%t LinearRecurrence[{2, 1, 1, 1}, {1, 1, 3, 8}, 31] (* _Michael De Vlieger_, Oct 28 2015 *)

%t nxt[{a_,b_,c_,d_}]:={b,c,d,2d+c+b+a}; NestList[nxt,{0,0,1,1},50][[All,1]] (* _Harvey P. Dale_, Mar 04 2022 *)

%o (Maxima)

%o a(n):=sum(sum((sum(binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,t-3*(k-j)-i),i,j,t-k+j),j,0,k))*binomial(-t+n+k-1,k-1),t,k,n),k,1,n);

%o (Magma) I:=[1,1,3,8]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Sep 20 2011

%o (PARI) x='x+O('x^30); Vec((1-x)/(1-2*x-x^2-x^3-x^4)) \\ _G. C. Greubel_, Dec 29 2017

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, May 05 2011

%E Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021