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A152464 Number of n-digit bouncy numbers in which every pair of adjacent digits are distinct. 1
0, 0, 525, 3105, 18939, 114381, 693129, 4195557, 25405586, 153820395, 931359050, 5639156409, 34143908573, 206733865761, 1251728824798, 7578945799704, 45888871327435, 277847147039527, 1682304127857000, 10185986079451152 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

where c is 2.32290643963522604128193759601...

Is c the result of some simple expression?

From Jon E. Schoenfield, Dec 16 2008: (Start)

We could define the recursive formula

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)

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).

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

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

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

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

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

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

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

LINKS

Table of n, a(n) for n=1..20.

FORMULA

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

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

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

u(1,i) = 1, and

d(1,i) = 1.

CROSSREFS

Cf. A043096, A152054.

Sequence in context: A281549 A260606 A181112 * A284961 A217742 A204692

Adjacent sequences:  A152461 A152462 A152463 * A152465 A152466 A152467

KEYWORD

base,nonn

AUTHOR

Jon E. Schoenfield, Dec 05 2008

EXTENSIONS

Correction to formula for odd negative n by Jon E. Schoenfield, Dec 22 2008

STATUS

approved

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Last modified February 17 02:33 EST 2019. Contains 320200 sequences. (Running on oeis4.)