A007689 - OEIS

A007689

a(n) = 2^n + 3^n.
(Formerly M1444)

2, 5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697, 847322163875

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

REFERENCES

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 14.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015-2017.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 169

Index entries for linear recurrences with constant coefficients, signature (5,-6).

FORMULA

E.g.f.: exp(2*x)*(1+exp(x)).

G.f.: (2-5*x)/((1-2*x)*(1-3*x)).

a(n) = 5*a(n-1) - 6*a(n-2).

Sum_{j=0..n-1} a(j) = (1/2)*(3^n - 1) + (2^n - 1). [Jolley] - Gary W. Adamson, Dec 20 2006

Equals double binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008

If p[i] = Fibonacci(2i-5) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010

a(n) = 2*a(n-1) + 3^(n-1), with a(0)=2. - Vincenzo Librandi, Nov 18 2010

a(n) = A001550(n) - 1 = A000079(n) + A000244(n). - Reinhard Zumkeller, Mar 01 2012

MAPLE

A007689:=n->2^n + 3^n: seq(A007689(n), n=0..50); # Wesley Ivan Hurt, Jan 24 2017

MATHEMATICA

Table[2^n + 3^n, {n, 0, 25}]

a=2; Numerator[Table[a=2*a-((a+1)/2), {n, 0, 7!}]] (*10 times (or more) faster for large numbers.*) (* Vladimir Joseph Stephan Orlovsky, Apr 19 2010 *)

LinearRecurrence[{5, -6}, {2, 5}, 30] (* nearly 20 times faster than the above program for large numbers. *) (* Harvey P. Dale, Oct 20 2013 *)

PROG

(Sage) [lucas_number2(n, 5, 6)for n in range(0, 27)] # Zerinvary Lajos, Jul 08 2008

(PARI) a(n)=2^n+3^n \\ Charles R Greathouse IV, Jun 15 2011

(Haskell)

a007689 n = a000079 n + a000244 n -- Reinhard Zumkeller, Apr 28 2013

(Magma) [2^n+3^n: n in [0..30]]; // G. C. Greubel, Mar 11 2023

CROSSREFS

For odd-indexed members divided by 5 see A096951.

Binomial transform of A000051.

Cf. A000079, A000244, A001550, A063376, A063481.

Cf. A074600 - A074624, A082101 (primes).