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A152464 Number of n-digit bouncy numbers in which every pair of adjacent digits are distinct. 2

%I

%S 0,0,525,3105,18939,114381,693129,4195557,25405586,153820395,

%T 931359050,5639156409,34143908573,206733865761,1251728824798,

%U 7578945799704,45888871327435,277847147039527,1682304127857000,10185986079451152

%N Number of n-digit bouncy numbers in which every pair of adjacent digits are distinct.

%C We might call such numbers "strictly bouncy numbers"; they exclude most n-digit "bouncy numbers" (cf. A152054) for n >= 4.

%C As n increases, a(n) approaches c/(2*cos(Pi*9/19))^n,

%C where c is 2.32290643963522604128193759601...

%C Is c the result of some simple expression?

%C From _Jon E. Schoenfield_, Dec 16 2008: (Start)

%C We could define the recursive formula

%C f(n) = 5*f(n-1) + 10*f(n-2) - 20*f(n-3) - 15*f(n-4) + 21*f(n-5) + 7*f(n-6) - 8*f(n-7) - f(n-8) + f(n-9)

%C and use a(n)=f(n) for n > 2 (a(n)=0 otherwise). Working backwards, given the terms f(11)=a(11) down through f(3)=a(3), the recursive formula would yield f(2)=81, f(1)=17 and f(0)=1, followed by the values 2, -1, 2, -2, 4, -5, 10, -14, 28, -42, 84, -132, etc., for negative values of n; these values are negative Catalan numbers for even n and twice (positive) Catalan numbers for odd n, down to f(-16).

%C The above results apply for numbers in base 10. In general, for base m+1 (so that the largest possible value for a digit is m), we can write

%C a(n) = f(n) for n > 2, 0 otherwise, where

%C f(n) = Sum_{j=1..m} (-1)^floor((j-1)/2)*binomial(floor((m+j)/2),j)*f(n-j) for n > 2,

%C f(2) = m^2, f(1) = 2*m - 1, f(0)=1,

%C f(n) = 2*Catalan((-1-n)/2) for odd n, 2 - 2m < n < 0 and

%C f(n) = -Catalan(-n/2) for even n, 2 - 2m <= n < 0.

%C (The expressions for n < 0 work more than far enough down to give enough terms to begin generating f(3), f(4), etc.) (End)

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,10,-20,-15,21,7,-8,-1,1).

%F a(n) = Sum_{i=1..9} (u(n,i) + d(n,i)) for n > 2 (0 otherwise), where

%F u(n,i) = Sum_{j=i+1..9} d(n-1,j) for n > 1,

%F d(n,i) = Sum_{j=0..i-1} u(n-1,j) for n > 1,

%F u(1,i) = 1, and

%F d(1,i) = 1.

%Y Cf. A043096, A152054.

%K base,nonn

%O 1,3

%A _Jon E. Schoenfield_, Dec 05 2008

%E Correction to formula for odd negative n by _Jon E. Schoenfield_, Dec 22 2008

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Last modified September 20 22:45 EDT 2021. Contains 347596 sequences. (Running on oeis4.)