A104720 Expansion of 1/((1-x)(1-x^2)(1-10x)).
1, 11, 112, 1122, 11223, 112233, 1122334, 11223344, 112233445, 1122334455, 11223344556, 112233445566, 1122334455667, 11223344556677, 112233445566778, 1122334455667788, 11223344556677889, 112233445566778899, 1122334455667789000, 11223344556677890010, 112233445566778900111, 1122334455667789001121
Offset: 0
Examples
From _Seiichi Manyama_, Sep 29 2018: (Start) 1 * 8 + 0 = 8; 11 * 8 + 1 = 89; 112 * 8 + 1 = 897; 1122 * 8 + 2 = 8978; 11223 * 8 + 2 = 89786; 112233 * 8 + 3 = 897867; 1122334 * 8 + 3 = 8978675; 11223344 * 8 + 4 = 89786756; 112233445 * 8 + 4 = 897867564; 1122334455 * 8 + 5 = 8978675645; 11223344556 * 8 + 5 = 89786756453; 112233445566 * 8 + 6 = 897867564534; 1122334455667 * 8 + 6 = 8978675645342; 11223344556677 * 8 + 7 = 89786756453423; 112233445566778 * 8 + 7 = 897767564534231; 1122334455667788 * 8 + 8 = 8978675645342312; 11223344556677889 * 8 + 8 = 89786756453423120; 112233445566778899 * 8 + 9 = 897867564534231201. 1 * 9 + 1 = 10; 11 * 9 + 2 = 101; 112 * 9 + 2 = 1010; 1122 * 9 + 3 = 10101; 11223 * 9 + 3 = 101010; 112233 * 9 + 4 = 1010101; 1122334 * 9 + 4 = 10101010; 11223344 * 9 + 5 = 101010101; 112233445 * 9 + 5 = 1010101010; 1122334455 * 9 + 6 = 10101010101; 11223344556 * 9 + 6 = 101010101010; 112233445566 * 9 + 7 = 1010101010101; 1122334455667 * 9 + 7 = 10101010101010; 11223344556677 * 9 + 8 = 101010101010101; 112233445566778 * 9 + 8 = 1010101010101010; 1122334455667788 * 9 + 9 = 10101010101010101; 11223344556677889 * 9 + 9 = 101010101010101010; 112233445566778899 * 9 + 10 = 1010101010101010101. (End)
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..999
- Index entries for linear recurrences with constant coefficients, signature (11,-9,-11,10).
Programs
-
GAP
List([0..25],n->1000*10^n/891+(-1)^n/44-(18*n+47)/324); # Muniru A Asiru, Sep 29 2018
-
Maple
seq(coeff(series(((1-x)*(1-x^2)*(1-10*x))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 29 2018
-
Mathematica
a[n_]:=1000*10^n/891 + (-1)^n/44 - (18*n + 47)/324 ; Array[a,50,0] (* or *) a[n_]:=Floor[(2*10^(n + 3) - 99*n)/1782]; Array[a,50,0] (* Stefano Spezia, Sep 01 2018 *) LinearRecurrence[{11,-9,-11,10},{1,11,112,1122},30] (* Harvey P. Dale, Jun 20 2021 *)
Formula
a(n) = 1000*10^n/891 + (-1)^n/44 - (18n+47)/324.
a(n) = floor((2*10^(n+3) - 99n)/1782). - Hieronymus Fischer, Dec 05 2006
a(n) = 10*a(n-1) + (2*n + 3 + (-1)^n)/4, a(0)=1, a(1)=11. - Vincenzo Librandi, Mar 22 2011
Comments