A391163 Numbers whose final digit is at most 2.

0, 1, 2, 10, 11, 12, 20, 21, 22, 30, 31, 32, 40, 41, 42, 50, 51, 52, 60, 61, 62, 70, 71, 72, 80, 81, 82, 90, 91, 92, 100, 101, 102, 110, 111, 112, 120, 121, 122, 130, 131, 132, 140, 141, 142, 150, 151, 152, 160, 161, 162, 170, 171, 172, 180, 181, 182, 190, 191, 192, 200

Offset

0,3

Comments

Numbers that are {0, 1, 2} mod 10.

Union of A008592, A017281, and A017293.

In baseball, a pitcher's length of time pitching is often measured by number of outs recorded; groups of three outs are separated by the "decimal" point of a(n)/10 from any additional outs.

Links

Zhuorui He, Table of n, a(n) for n = 0..10000

Wikipedia, Innings pitched.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

a(n) = 10 * floor(n/3) + (n mod 3) = 10 * A002264(n) + A010872(n).

G.f.: x*(1 + x + 8*x^2)/((-1 + x)^2*(1 + x + x^2)).

a(n)= A054967(n) + 1 for n >= 1.

E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(10*x - 7) + 21*cos(sqrt(3)*x/2) + 7*sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Dec 05 2025

Example

A pitcher that has recorded 22 outs has pitched 7.1 innings, so a(22) = 71.

Mathematica

a[n_] := Total[QuotientRemainder[n, 3] * {10, 1}]; Array[a, 61, 0]
Select[Range[0, 200], Mod[#, 10]<3&] (* or *) LinearRecurrence[{1, 0, 1, -1}, {0, 1, 2, 10}, 100] (* Harvey P. Dale, Jan 27 2026 *)

Crossrefs

Cf. A002264, A010872.

Cf. A008592 (final digit at most 0), A197652 (at most 1), A390774 (at most 3), A293292 (at most 4), A391164 (at most 5), A272576 (at most 7), A001477 (at most 9).

Cf. A017281, A017293, A054967.

Sequence in context: A217072 A106518 A158304 * A007089 A159952 A136820

Adjacent sequences: A391160 A391161 A391162 * A391164 A391165 A391166

Keyword

nonn,easy,base

Author

Jason Bard, Dec 01 2025

Status

approved