OFFSET
0,2
COMMENTS
n-th difference of a(n), a(n-1), ..., a(0) is A001018(n-1) for n >= 1.
Also, the cogrowth sequence of the 16-element group D4 X C2 = <S,T,U | S^4, T^2, U^2, (ST)^2, [S,T], [U,T]>. - Sean A. Irvine, Nov 10 2024
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).
FORMULA
a(n) = (9^n + 7)/8. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 06 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 10*a(n-1) - 9*a(n-2) for n > 1.
G.f.: (1 - 8*x)/(1 - 10*x + 9*x^2). (End)
a(n) = 9*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(8*x) + 7)/8. - Elmo R. Oliveira, Aug 29 2024
MAPLE
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=9*a[n-1]+1 od: seq(a[n]+1, n=0..17); # Zerinvary Lajos, Mar 20 2008
MATHEMATICA
a = {1}; ZZ = 1; Do[ZZ = ZZ + 3^(2x); AppendTo[a, ZZ], {x, 0, 40}]; a (* Zerinvary Lajos, Apr 03 2007 *)
(9^Range[0, 40] +7)/8 (* G. C. Greubel, Jan 12 2025 *)
PROG
(Magma) [(9^n +7)/8: n in [0..40]]; // G. C. Greubel, Jan 12 2025
(Python)
def A047854(n): return (pow(9, n) +7)//8
print([A047854(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
a(18)-a(22) from Elmo R. Oliveira, Aug 29 2024
STATUS
approved