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A190139
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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.
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1
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1, 1, 3, 8, 21, 54, 140, 363, 941, 2439, 6322, 16387, 42476, 110100, 285385, 739733, 1917427, 4970072, 12882689, 33392610, 86555408, 224356187, 581543081, 1507390367, 3907235410, 10127760455, 26251689768, 68045765768, 176378217169, 457181650329, 1185038973363
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: (1-x)/(1-2*x-x^2-x^3-x^4).
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.
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MATHEMATICA
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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 *)
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 *)
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PROG
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(Maxima)
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);
(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
(PARI) x='x+O('x^30); Vec((1-x)/(1-2*x-x^2-x^3-x^4)) \\ G. C. Greubel, Dec 29 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
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STATUS
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approved
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