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%I #10 Feb 15 2024 08:52:46
%S 1,10,96,924,8892,85572,823500,7924932,76265388,733938084,7063035084,
%T 67970944260,654116708844,6294876045156,60578584659468,
%U 582976518206148,5610260171812140,53990200655546148,519573366930788172,5000101506310370436,48118353758378062956
%N Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 8.
%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 (9,6).
%F a(n) = 9*a(n-1) + 6*a(n-2), a(0)=1, a(1)=10.
%F G.f.: (-1 - x)/(-1 + 9*x + 6*x^2).
%F a(n) = ((1 - 11/sqrt(105))/2)*((9 - sqrt(105))/2)^n + ((1 + 11/sqrt(105))/2)*((9 + sqrt(105))/2)^n.
%t LinearRecurrence[{9, 6}, {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)+6*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
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