OFFSET
3,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..1003
Kevin Beanland, Dmitriy Gorovoy, Jȩdrzej Hodor, and Daniil Homza, Counting Unions of Schreier Sets, arXiv:2211.01049 [math.CO], 2022. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1).
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 ...
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
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