A094626 Expansion of x*(1+x)/((1-x)*(1-10*x^2)).

0, 1, 2, 12, 22, 122, 222, 1222, 2222, 12222, 22222, 122222, 222222, 1222222, 2222222, 12222222, 22222222, 122222222, 222222222, 1222222222, 2222222222, 12222222222, 22222222222, 122222222222, 222222222222, 1222222222222, 2222222222222, 12222222222222

Offset: 0

Author

Paul Barry, May 15 2004

Keywords

• easy
• nonn
• base

Comments

Previous name: Sequence whose n-th term digits sum to n.

a(n) is the smallest integer with digits from {0,1,2} having digit sum n. Namely the base-10 reading of the ternary string of A062318. - Jason Kimberley, Nov 01 2011

a(n) is the Moore lower bound on the order of an (11,n)-cage. - Jason Kimberley, Oct 18 2011

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• G. Royle, Cages of higher valency
• Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Crossrefs

Cf. A094623, A094624.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), this sequence (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 01 2011

Programs

Mathematica

LinearRecurrence[{1, 10, -10}, {0, 1, 2}, 30] (* Paolo Xausa, Feb 21 2024 *)

PARI

concat(0, Vec(x*(1+x)/((1-x)*(1-10*x^2)) + O(x^30))) \\ Colin Barker, Mar 17 2017

Formula

G.f.: x*(1+x)/((1-x)*(1-10*x^2)).

a(n) = 10^(n/2)*(11*sqrt(10)/180 + 1/9 - (11*sqrt(10)/180 - 1/9)*(-1)^n) - 2/9.

From Colin Barker, Mar 17 2017: (Start)

a(n) = 2*(10^(n/2) - 1)/9 for n even.

a(n) = (11*10^((n-1)/2) - 2)/9 for n odd. (End)

E.g.f.: (20*(cosh(sqrt(10)*x) - cosh(x) - sinh(x)) + 11*sqrt(10)*sinh(sqrt(10)*x))/90. - Stefano Spezia, Apr 09 2022