A015528 a(n) = 3*a(n-1) + 10*a(n-2).

0, 1, 3, 19, 87, 451, 2223, 11179, 55767, 279091, 1394943, 6975739, 34876647, 174387331, 871928463, 4359658699, 21798260727, 108991369171, 544956714783, 2724783836059, 13623918656007, 68119594328611, 340597969545903, 1702989851923819, 8514949251230487, 42574746272929651

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

The ratio a(n+1)/a(n) converges to 5 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014.
• Index entries for linear recurrences with constant coefficients, signature (3,10).

Programs

Magma

[5^n/7-(-2)^n/7: n in [0..30]]; // Vincenzo Librandi, Aug 23 2011

Mathematica

Join[{a=0,b=1},Table[c=3*b+10*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{3,10},{0,1},30] (* or *) CoefficientList[Series[x/(1-x (10x+3)),{x,0,29}],x] (* Harvey P. Dale, Jan 27 2012 *)

PARI

for(n=0,30, print1((5^n - (-2)^n)/7, ", ")) \\ G. C. Greubel, Jan 01 2018

Sage

[lucas_number1(n,3,-10) for n in range(0, 23)]# Zerinvary Lajos, Apr 22 2009

Formula

a(n) = 3*a(n-1) + 10*a(n-2).

a(n) = (5^n - (-2)^n)/7. Binomial transform is A015540. - Paul Barry, Feb 07 2004

G.f.: x/(1 - x*(10*x+3)). - Harvey P. Dale, Jan 27 2012

E.g.f.: exp(-2*x)*(exp(7*x) - 1)/7. - Elmo R. Oliveira, Apr 02 2025