A287830 - OEIS

%I #14 Feb 15 2024 08:52:29

%S 1,10,94,886,8350,78694,741646,6989590,65872894,620814406,5850821230,

%T 55140648694,519669123166,4897584703270,46156938822094,

%U 435002788211926,4099652849195710,38636886795609094,364130592557264686,3431722880197818550,32342028292009425694

%N Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 7.

%C In general, the number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 5+k for k in {0,1,2,3,4} is given by a(n) = 9*a(n-1) + 2*k*a(n-2), a(0)=1, a(1)=10.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,4).

%F a(n) = 9*a(n-1) + 4*a(n-2), a(0)=1, a(1)=10.

%F G.f.: (-1 - x)/(-1 + 9*x + 4*x^2).

%F a(n) = ((1 - 11/sqrt(97))/2)*((9 - sqrt(97))/2)^n + ((1 + 11/sqrt(97))/2)*((9 + sqrt(97))/2)^n.

%F a(n) = A015580(n)+A015580(n+1). - _R. J. Mathar_, Oct 20 2019

%t LinearRecurrence[{9, 4}, {1, 10}, 30]

%o (Python)

%o def a(n):

%o .if n in [0, 1]:

%o ..return [1, 10][n]

%o .return 9*a(n-1)+4*a(n-2)

%Y Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

%K nonn,easy

%O 0,2

%A _David Nacin_, Jun 02 2017