A276849 a(0) = 5, a(1) = 2; for n>1, a(n) = 2*a(n-1) + a(n-2).

5, 2, 9, 20, 49, 118, 285, 688, 1661, 4010, 9681, 23372, 56425, 136222, 328869, 793960, 1916789, 4627538, 11171865, 26971268, 65114401, 157200070, 379514541, 916229152, 2211972845, 5340174842, 12892322529, 31124819900, 75141962329, 181408744558, 437959451445

Offset

0,1

Comments

After the first two terms, there are no primes in this sequence. In fact:

a(12*k) is divisible by 5,

a(12*k+1) is divisible by 2,

a(12*k+2) is divisible by 3,

a(12*k+3) is divisible by 2,

a(12*k+4) is divisible by 7,

a(12*k+5) is divisible by 2,

a(12*k+6) is divisible by 3,

a(12*k+7) is divisible by 2,

a(12*k+8) is divisible by 11,

a(12*k+9) is divisible by 2,

a(12*k+10) is divisible by 3,

a(12*k+11) is divisible by 2.

Therefore, every term is divisible by 2, 3, 5, 7 or 11.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1).

Formula

From Colin Barker, Oct 11 2016: (Start)

G.f.: (5-8*x) / (1-2*x-x^2).

a(n) = ((1 + sqrt(2))^n*(-3 + 5*sqrt(2)) + (1 - sqrt(2))^n*(3 + 5*sqrt(2)))/(2*sqrt(2)). (End)

Mathematica

CoefficientList[Series[(5 - 8*x)/(1 - 2*x - x^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 11 2016 *)
a[0] = 5; a[1] = 2; a[n_] := a[n] = 2 a[n - 1] + a[n - 2]; Array[a, 31, 0] (* or *) LinearRecurrence[{2, 1}, {5, 2}, 31] (* Robert G. Wilson v, Oct 12 2016 *)

Prog

(PARI) lista(n)=n++; my(v=vector(max(2, n))); v[1]=5; v[2]=2; for(i=3, n, v[i]=2*v[i-1]+v[i-2]); v \\ David A. Corneth, Oct 11 2016
(PARI) Vec((5-8*x)/(1-2*x-x^2) + O(x^40)) \\ Colin Barker, Oct 11 2016

Crossrefs

Cf. A277369.

Sequence in context: A274930 A080379 A257513 * A367210 A127098 A127097

Adjacent sequences: A276846 A276847 A276848 * A276850 A276851 A276852

Keyword

nonn,easy

Author

Bobby Jacobs, Oct 11 2016

Extensions

More terms from David A. Corneth, Oct 11 2016

Status

approved