A255821 Numbers of words on {0,1,...,36} having no isolated zeros.

1, 36, 1297, 46729, 1683577, 60656797, 2185374961, 78735837637, 2836736138665, 102203420474269, 3682238546710945, 132665625592223221, 4779746882367738841, 172207232713967895181, 6204372685172893559377, 223534399861459456068709

Offset

0,2

Comments

The number p_n = a(n)/37^n equals the probability that in n trials in single zero (European) Roulette zero will not appear isolated. For example, p_10 is approximately 0.021.

Links

Colin Barker, Table of n, a(n) for n = 0..642

Index entries for linear recurrences with constant coefficients, signature (37,-36,36).

Formula

G.f.: -(x^2 - x + 1)/(36*x^3 - 36*x^2 + 37*x - 1). - Colin Barker, Mar 09 2015

a(n) = 37*a(n-1) - 36*a(n-2) + 36*a(n-3). - G. C. Greubel, Jun 02 2016

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 36,  a[2]== 1297, a[n] == 37 a[n - 1] - 36 a[n - 2] + 36 a[n - 3]}, a[n], {n, 0, 15}]
LinearRecurrence[{37, -36, 36}, {1, 36, 1297}, 100] (* G. C. Greubel, Jun 02 2016 *)

Prog

(PARI) Vec(-(x^2-x+1)/(36*x^3-36*x^2+37*x-1) + O(x^100)) \\ Colin Barker, Mar 09 2015

Crossrefs

Cf. A255116, A255118, A254658, A254660, A255633, A255630.

Sequence in context: A300357 A009980 A041613 * A209042 A283729 A203333

Adjacent sequences: A255818 A255819 A255820 * A255822 A255823 A255824

Keyword

nonn,easy

Author

Milan Janjic, Mar 07 2015

Status

approved