A201630 a(n) = a(n-1) + 2*a(n-2) with n>1, a(0)=2, a(1)=7.

2, 7, 11, 25, 47, 97, 191, 385, 767, 1537, 3071, 6145, 12287, 24577, 49151, 98305, 196607, 393217, 786431, 1572865, 3145727, 6291457, 12582911, 25165825, 50331647, 100663297, 201326591, 402653185, 805306367, 1610612737, 3221225471, 6442450945, 12884901887

Offset: 0

Author

Bruno Berselli, Dec 03 2011

Keywords

• nonn
• easy

References

• B. Satyanarayana and K. S. Prasad, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd. (Eastern Economy Edition), 2009, p. 73 (problem 3.3).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A001045, A005010, A042948, A097581.

Programs

Magma

[n le 2 select 5*n-3 else Self(n-1)+2*Self(n-2): n in [1..33]];

Mathematica

LinearRecurrence[{1, 2}, {2,7}, 33]

Maxima

a[0]:2$ a[1]:7$ a[n]:=a[n-1]+2*a[n-2]$ makelist(a[n], n, 0, 32);

PARI

v=vector(33); v[1]=2; v[2]=7; for(i=3, #v, v[i]=v[i-1]+2*v[i-2]); v

SageMath

def A201630(n): return 3*2**n - (-1)**n
print([A201630(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

Formula

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

a(n) = 3*2^n - (-1)^n.

a(n) = 7 + 2*Sum_{i=0..n-2} a(i), for n>0.

a(n) = A097581(A042948(n+1)).

a(n+2) - a(n) = a(n+1) + a(n) = A005010(n).

E.g.f.: 3*exp(2*x) - exp(-x). - G. C. Greubel, Feb 07 2025