OFFSET
1,3
COMMENTS
A bouncy number has digits that are neither monotonically increasing nor decreasing from left to right.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..200
Project Euler, Non-bouncy numbers Problem 113
Index entries for linear recurrences with constant coefficients, signature (21,-165,715,-1980,3762,-5082,4950,-3465,1705,-561,111,-10).
FORMULA
a(1) = a(2) = 0. For n > 2, a(n) = 10^n - binomial(n+9, n) + 10*n - Sum_{i=1..n} binomial(i+9, i) = 10^n - binomial(n+10, 10) - binomial(n+9, 9) + 10*n + 1. [Corrected by Altug Alkan, Sep 28 2018]
From Chai Wah Wu, Jul 28 2023: (Start)
a(n) = 21*a(n-1) - 165*a(n-2) + 715*a(n-3) - 1980*a(n-4) + 3762*a(n-5) - 5082*a(n-6) + 4950*a(n-7) - 3465*a(n-8) + 1705*a(n-9) - 561*a(n-10) + 111*a(n-11) - 10*a(n-12) for n > 12.
G.f.: x^3*(89*x^8 - 798*x^7 + 3175*x^6 - 7350*x^5 + 10890*x^4 - 10668*x^3 + 6846*x^2 - 2700*x + 525)/((x - 1)^11*(10*x - 1)). (End)
EXAMPLE
MAPLE
a:=n->`if`(n<=2, 0, 10^n-(binomial(n+10, 10)+binomial(n+9, 9)-1-10*n)); seq(a(n), n=1..25); # Muniru A Asiru, Sep 28 2018
MATHEMATICA
a[n_]:= 10^n - Binomial[n+10, 10] - Binomial[n+9, 9] + 10*n + 1; Array[a, 25] (* Stefano Spezia, Sep 29 2018 *)
PROG
(PARI) a(n) = if(n<3, 0, 10^n-(binomial(n+10, 10)+binomial(n+9, 9)-1-10*n)); \\ Altug Alkan, Sep 27 2018
(GAP) a:=[0, 0];; for n in [3..25] do a[n]:=10^n-(Binomial(n+10, 10)+Binomial(n+9, 9)-1-10*n); od; a; # Muniru A Asiru, Sep 28 2018
(Magma) [n le 2 select 0 else (10^n - Binomial(n+10, 10) - Binomial(n+9, 9) + 10*n + 1): n in [1..25]]; // G. C. Greubel, Nov 24 2018
(Sage)
def A204692(n):
if n <= 2:
return 0
else:
return (10^n - binomial(n+10, 10) - binomial(n+9, 9) + 10*n + 1)
[A204692(n) for n in (1..25)] # G. C. Greubel, Nov 24 2018
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Thomas Quirk, Jan 18 2012
EXTENSIONS
Corrected and extended by Altug Alkan, Sep 27 2018
STATUS
approved