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A059633
Expansion of g.f. x^3/(1 - 2*x + x^3 - x^4).
9
1, 2, 4, 7, 13, 24, 45, 84, 157, 293, 547, 1021, 1906, 3558, 6642, 12399, 23146, 43208, 80659, 150571, 281080, 524709, 979506, 1828503, 3413377, 6371957, 11894917, 22204960, 41451340, 77379720, 144449397, 269652414
OFFSET
3,2
LINKS
Kevin Beanland, Dmitriy Gorovoy, Jȩdrzej Hodor, and Daniil Homza, Counting Unions of Schreier Sets, arXiv:2211.01049 [math.CO], 2022. See p. 4.
FORMULA
a(n) = 2*a(n-1) - a(n-3) + a(n-4).
Recurrence equations (A059633 is L(n)): I(n+1) = I(n) + J(n) + L(n); J(n+1) = I(n); K(n+1) = J(n) + K(n); L(n+1) = K(n); M(n+1) = L(n) + 2M(n); initial conditions: I(0) = 1; J(0) = 0; K(0) = 0; L(0) = 0; M(0) = 0. Values for n = 0 1 2 3 4 5 6 7 8 ...: I(n) = 1 1 2 3 6 11 21 39 73 ... J(n) = 0 1 1 2 3 6 11 21 39 ... K(n) = 0 0 1 2 4 7 13 24 45 ... L(n) = 0 0 0 1 2 4 7 13 24 ... M(n) = 0 0 0 0 1 4 12 31 75 ...
a(n) = A049856(n+2) - A049856(n+1) - A049856(n) + A049856(n-1).
For n >= 2, a(n+1) = Sum_{i=0..n} Fibonacci(i)*binomial(n-i, i). - Benoit Cloitre, Sep 21 2004
a(n) = Sum_{k=0..n+1} C(k+1, n-k+1)*Fibonacci(n-k+1) [offset 0]. - Paul Barry, Feb 23 2005
MAPLE
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card > 1), U=Sequence(Z, card >1)}, unlabeled]: seq(count(SeqSeqSeqL, size=j), j=4..35); # Zerinvary Lajos, Apr 04 2009
MATHEMATICA
LinearRecurrence[{2, 0, -1, 1}, {1, 2, 4, 7}, 40] (* Harvey P. Dale, Dec 25 2022 *)
PROG
(Magma) I:=[1, 2, 4, 7]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Apr 13 2023
(SageMath)
@CachedFunction
def a(n): # a = A059633
if (n<4): return (0, 0, 0, 1, 2, 4, 7)[n]
else: return 2*a(n-1) - a(n-3) + a(n-4)
[a(n) for n in range(3, 51)] # G. C. Greubel, Apr 13 2023
CROSSREFS
I and J are A049856 while K and L are A059633 (with some offsets).
Sequence in context: A102111 A224704 A265826 * A088353 A192654 A260668
KEYWORD
nonn,easy
AUTHOR
James R. FitzSimons (cherry(AT)neta.com), Feb 19 2001
EXTENSIONS
Comments and more terms from Henry Bottomley, Feb 21 2001
New description from Vladeta Jovovic, Jan 17 2004
STATUS
approved