OFFSET
0,3
COMMENTS
If the initial 1 is omitted, this is 2^n mod 9. - N. J. A. Sloane
From Cino Hilliard, Dec 31 2004: (Start)
Except for the initial term, also the digital root of 11^n.
Except for the initial term, also the decimal expansion of 125/1001.
Except for the initial term, also the digital root of 2^n. (End)
Aside from the first term, periodic with period 6. - Charles R Greathouse IV, Nov 29 2011
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).
FORMULA
a(n) = digital root of 2^(n-1) in base 10 = 2^(n-1) (mod 9). - Olivier Gérard, Jun 06 2001
For n > 0: a(n+6) = a(n) and a(n) = A007612(n+1) - A007612(n) = A010888(A007612(n)). - Reinhard Zumkeller, Feb 27 2006
a(n) = (9 + cos(n*Pi) - 4*sqrt(3)*sin(n*Pi/3))/2 for n > 0 with a(0)=1. - Wesley Ivan Hurt, Oct 04 2018
From Stefano Spezia, Jun 27 2022: (Start)
O.g.f.: (1 + x^2 + 3*x^3 + 4*x^4)/((1 - x)*(1 + x)*(1 - x + x^2)).
E.g.f.: 5*cosh(x) - 2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) + 4*(sinh(x) - 1). (End)
EXAMPLE
1 + 1 + 2 + 4 + 8 + 7 + 5 = 28 -> 2 + 8 = 10 -> a(7) = 1.
MATHEMATICA
a[n_] := PowerMod[2, n-1, 9]; a[0] = 1; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 29 2011 *)
Join[{1}, LinearRecurrence[{1, 0, -1, 1}, {1, 2, 4, 8}, 110]] (* or *) Join[{1}, PowerMod[2, Range[110], 9]] (* Harvey P. Dale, Nov 24 2014 *)
PROG
(Sage) [power_mod(2, n, 9)for n in range(0, 105)] # Zerinvary Lajos, Nov 03 2009
(PARI) a(n)=if(n, [5, 1, 2, 4, 8, 7][n%6+1], 1) \\ Charles R Greathouse IV, Nov 29 2011
CROSSREFS
KEYWORD
base,nonn,nice,easy
AUTHOR
Amela2(AT)aol.com
EXTENSIONS
More terms from Cino Hilliard, Dec 31 2004
STATUS
approved