A277209 Partial sums of repdigit numbers (A010785).

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 56, 78, 111, 155, 210, 276, 353, 441, 540, 651, 873, 1206, 1650, 2205, 2871, 3648, 4536, 5535, 6646, 8868, 12201, 16645, 22200, 28866, 36643, 45531, 55530, 66641, 88863, 122196, 166640, 222195, 288861, 366638, 455526, 555525, 666636, 888858, 1222191, 1666635, 2222190

Offset: 0

Author

Ilya Gutkovskiy, Oct 05 2016

Keywords

• nonn
• base
• easy

Comments

More generally, the ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1) is (Sum_{m = 1..(k-1)} m*x^m)/((1 - x)*(1 - x^(k-1))*(1 - k*x^(k-1))).

Examples

			a(0)=0;
a(1)=0+1=1;
a(2)=0+1+2=3;
a(3)=0+1+2+3=6;
...
a(10)=0+1+2+3+4+5+6+7+8+9+11=56;
a(11)=0+1+2+3+4+5+6+7+8+9+11+22=78;
a(12)=0+1+2+3+4+5+6+7+8+9+11+22+33=111, etc.
		

Links

• Eric Weisstein's World of Mathematics, Repdigit
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,11,-11,0,0,0,0,0,0,0,-10,10).

Crossrefs

Cf. A000217, A010785, A027828, A046489.

Programs

Mathematica

CoefficientList[Series[x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5 + 7 x^6 + 8 x^7 + 9 x^8)/((1 - x) (1 - 10 x^9) (1 - x^9)), {x, 0, 50}], x]

Formula

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8)/((1 - x)*(1 - x^9)*(1 - 10*x^9)).

a(n) = A000217(n) for n < 10.

a(n) = A046489(n-1) for n < 19.