OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..500
Pierre-Alain Sallard, Integer sequences A328348 and A328350 to A328356.
Index entries for linear recurrences with constant coefficients, signature (100,-909,810).
FORMULA
a(n) = (80*90^n - 89*9^n + 9) * (5/712).
a(n) = 91*a(n-1) - 90*a(n-2) + 45*9^(n-1) for n > 1.
G.f.: 45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)). - Colin Barker, Dec 10 2019
From Elmo R. Oliveira, Jul 08 2026: (Start)
a(n) = 100*a(n-1) - 909*a(n-2) + 810*a(n-3).
E.g.f.: (5/712)*exp(x)*(9 - 89*exp(8*x) + 80*exp(89*x)). (End)
EXAMPLE
For n=2, the sum of all integers from 1 to 99 except those containing a zero (i.e., except multiples of 10: 10, 20, ..., 90) is equal to a(2) = 4500.
For n=3, the sum of all integers from 1 to 999 except those containing a zero is equal to a(3) = 409095.
MATHEMATICA
(* Alternative: *)
LinearRecurrence[{100, -909, 810}, {0, 45, 4500}, 20] (* Paolo Xausa, May 11 2026 *)
PROG
(Python) [(80*90**n-89*9**n+9)*5//712 for n in range(20)]
(PARI) concat(0, Vec(45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)) + O(x^15))) \\ Colin Barker, Dec 10 2019
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Pierre-Alain Sallard, Dec 10 2019
STATUS
approved