OFFSET
1,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,19,-19,-90,90).
FORMULA
G.f.: (9*x^2*(x+1))/((1-x)*(1 - 9*x^2)*(1 - 10*x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
From G. C. Greubel, Oct 27 2016: (Start)
a(n) = a(n-1) + 19*a(n-2) - 19*a(n-3) - 90*a(n-4) + 90*a(n-5).
a(n) = (1/(4*sqrt(10)))*( 4*sqrt(10)*(1 + (-1)^n)*(10)^(n/2) + 22*(1 - (-1)^n)*(10)^(n/2) + sqrt(10)*(1 + ((-1)^n - 4)*3^(n + 1)) ).
E.g.f.: (1/(4*sqrt(10)))*( sqrt(10)*(3*exp(-3*x) + exp(x) -12*exp(3*x)) + 44*sinh(sqrt(10)*x) + 8*sqrt(10)*cosh(sqrt(10)*x)).
a(2*n) = (1/4)*(1 + 8*(10)^n - 9^(1 + n)), n>=1.
a(2*n+1) = (1/20)*(5 + 22*(10)^(n+1) - 25*9^(n+1)), n>=0. (End)
EXAMPLE
Up to 10^4 we find 18 numbers -> 101, 202, ..., 909, 1001, 2002, ... and 9009.
MATHEMATICA
LinearRecurrence[{1, 19, -19, -90, 90}, {0, 0, 9, 18, 189}, 25] (* G. C. Greubel, Oct 27 2016 *)
(* Alternative: *)
Table[If[OddQ[n], (5 + 22*(10)^((n + 1)/2) - 25*9^((n + 1)/2))/20, (1 + 8*(10)^(n/2) - 9^((n/2) + 1))/4], {n, 1, 10}] (* G. C. Greubel, Oct 27 2016 *)
PROG
(Magma) [IsOdd(n) select (5+22*(10)^((n+1) div 2)-25*9^((n+1) div 2)) div 20 else (1+8*(10)^(n div 2)-9^((n div 2)+1)) div 4:n in [1..30]]; // Vincenzo Librandi, Oct 29 2016
(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 90, -90, -19, 19, 1]^(n-1)*[0; 0; 9; 18; 189])[1, 1] \\ Charles R Greathouse IV, May 29 2026
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Patrick De Geest, Aug 15 1999
EXTENSIONS
More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999
Corrected by T. D. Noe, Nov 08 2006
STATUS
approved