A094627 Expansion of (1+x)^2/((1-x)*(1-10*x^2)).

1, 3, 14, 34, 144, 344, 1444, 3444, 14444, 34444, 144444, 344444, 1444444, 3444444, 14444444, 34444444, 144444444, 344444444, 1444444444, 3444444444, 14444444444, 34444444444, 144444444444, 344444444444, 1444444444444

Offset: 0

Author

Paul Barry, May 15 2004

Keywords

• easy
• nonn

Comments

The digital sum of the n-th term is 2n+1.

a(n) = floor(10^floor(n/2)*(2-(-1)^n+4/9)) = 1,3,14,34,144,344,... (i.e. 2-(-1)^n = 1 for even n, 3 for odd n, followed by floor(n/2) digits '4'.) - M. F. Hasler

Examples

			(x^2 + 2*x + 1)/(10*x^3 - 10*x^2 - x + 1) = 1 + 3*x + 14*x^2 + 34*x^3 + 144*x^4 + 344*x^5 + 1444*x^6 + 3444*x^7 + 14444*x^8 + ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Crossrefs

Cf. A094626.

Programs

Maple

A011557aer := proc(n) if type(n,'odd') then 0 ; else 10^(n/2) ; end if; end proc:
A094627 := proc(n) (13*A011557aer(n)+31*A011557aer(n-1)-4)/9 ; end proc:
seq(A094627(n),n=0..10) ; # R. J. Mathar, Nov 16 2010

Mathematica

sr[n_,nn_]:=Table[FromDigits[PadRight[{n},i,4]],{i,nn}]; With[{nn=20}, Sort[ Join[ sr[ 1,nn],sr[3,nn]]]] (* Harvey P. Dale, May 25 2014 *)

Formula

a(n) = 10^(n/2)*( 31*sqrt(10)/180 +13/18 -(31*sqrt(10)/180-13/18)*(-1)^n )-4/9.

a(n) = (13*b(n)+31*b(n-1)-4)/9 with b(n) = 1,0,10,0,100,0,1000,.. (aerated A011557) [R. J. Mathar, Nov 26 2010]

Extensions

Swapped the generic comment and the specific definition; added Maple prog.