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A118898
Number of binary sequences of length n containing exactly one subsequence 0000.
3
0, 0, 0, 0, 1, 2, 5, 12, 28, 62, 136, 294, 628, 1328, 2787, 5810, 12043, 24840, 51016, 104380, 212848, 432732, 877400, 1774672, 3581605, 7213746, 14502449, 29106100, 58323844, 116702074, 233199000, 465405058, 927744428, 1847359520, 3674769991
OFFSET
0,6
COMMENTS
Column 1 of A118897.
LINKS
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
FORMULA
G.f.: z^4/(1-z-z^2-z^3-z^4)^2.
From Bobby Milazzo, Aug 30 2009: (Start)
a(1)=0,a(2)=0,a(3)=0,a(4)=1,a(5)=2,a(6)=5,a(7)=12,a(8)=28
a(n) = 2a(n-1)+a(n-2)-a(n-4)-4a(n-5)-3a(n-6)-2a(n-7)-a(n-8). (End)
EXAMPLE
a(6)=5 because we have 000010,000011,010000,100001 and 110000.
G.f. = x^4 + 2*x^5 + 5*x^6 + 12*x^7 + 28*x^8 + 62*x^9 + ... - Zerinvary Lajos, Jun 02 2009
MAPLE
g:=z^4/(1-z-z^2-z^3-z^4)^2: gser:=series(g, z=0, 40): seq(coeff(gser, z, n), n=0..37);
MATHEMATICA
RecurrenceTable[{a[1]==0, a[2]==0, a[3]==0, a[4]==1, a[5]==2, a[6]==5, a[7]==12, a[8]==28, a[n]==2a[n-1]+a[n-2]-a[n-4]-4a[n-5]-3a[n-6]-2a[n-7]-a[n-8]}, a, {n, 9, 50}] (* Bobby Milazzo, Aug 30 2009 *)
LinearRecurrence[{2, 1, 0, -1, -4, -3, -2, -1}, {0, 0, 0, 0, 1, 2, 5, 12}, 50] (* Harvey P. Dale, Aug 01 2012 *)
PROG
(Sage) taylor( mul(x/(1-x-x^2-x^3-x^4)^2 for i in range(1, 2)), x, 0, 31)# Zerinvary Lajos, Jun 02 2009
CROSSREFS
Cf. A118897.
Sequence in context: A171579 A228638 A202604 * A111586 A192657 A320590
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 04 2006
STATUS
approved