A229470 Positions of 2 in decimal expansion of 0.1231232331232332333..., whose definition is given below.

2, 5, 7, 11, 13, 16, 21, 23, 26, 30, 36, 38, 41, 45, 50, 57, 59, 62, 66, 71, 77, 85, 87, 90, 94, 99, 105, 112, 121, 123, 126, 130, 135, 141, 148, 156, 166, 168, 171, 175, 180, 186, 193, 201, 210, 221, 223, 226, 230, 235, 241, 248, 256, 265, 275, 287, 289, 292, 296, 301, 307, 314, 322, 331, 341, 352, 365, 367, 370, 374, 379, 385

Offset: 1

Author

Jiri Klepl, Sep 24 2013

Keywords

• easy
• nonn
• base

Comments

0.1231232331232332333... = Sum_{k>=0} 10^(-(k + 3)! / (3! * k!)) * (1 + 10 * Sum_{l=2..k+2} 10^(-(l^2 + l) / 2) * ((10^l - 1) / 3 - 10^(l - 1))).

Programs

PARI

a(n)=sum(k=0,n-1,1+k-binomial(round(sqrt(2+2*k)),2)+issquare(8*k+1)*(sqrtint(1+8*k)+1)/2) /* Ralf Stephan, Oct 09 2013 */

Formula

a((n^2+n+2m-2)/2) = (n^3+6n^2+3m^2+11n-3m+6)/6; n+2>=m>=2.

a(n) = Sum_{k=0..n-1} ( 1 + A002262(k) + A010054(k)*(sqrt(1+8*k)+1)/2 ).

Extensions

Formula corrected by Ralf Stephan, Oct 09 2013