A047854 a(n) = A047848(6, n).

1, 2, 11, 92, 821, 7382, 66431, 597872, 5380841, 48427562, 435848051, 3922632452, 35303692061, 317733228542, 2859599056871, 25736391511832, 231627523606481, 2084647712458322, 18761829412124891, 168856464709124012, 1519708182382116101, 13677373641439044902, 123096362772951404111

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

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 = . - 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).

Crossrefs

Cf. A001018, A047848.

Programs

Magma

[(9^n +7)/8: n in [0..40]]; // G. C. Greubel, Jan 12 2025

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 *)

Python

def A047854(n): return (pow(9,n) +7)//8
print([A047854(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

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

Extensions

a(18)-a(22) from Elmo R. Oliveira, Aug 29 2024