A268409 a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5.

3, 5, 26, 114, 508, 2260, 10056, 44744, 199088, 885840, 3941536, 17537824, 78034368, 347213120, 1544921216, 6874111104, 30586286848, 136093369600, 605546052096, 2694370947584, 11988575894528, 53343045473280, 237349333682176, 1056083425675264

Offset: 0

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

• nonn
• easy

Comments

In general, the ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m, is (k - (k*r - m)*x)/(1 - r*x - s*x^2). This recurrence gives the closed form b(n) = (2^(-n - 1)*((k*r - 2*m)*(r - sqrt(r^2 + 4*s))^n + (2*m - k*r)*(sqrt(r^2 + 4*s) + r)^n + k*sqrt(r^2 + 4*s)*(r - sqrt(r^2 + 4*s))^n + k*sqrt(r^2 + 4*s)*(sqrt(r^2 + 4*s) + r)^n))/sqrt(r^2 + 4*s).

Links

• Index entries for linear recurrences with constant coefficients, signature (4,2).

Crossrefs

Cf. A021001, A084059, A090017, A107979, A164549, A176213, A189741.

Programs

Magma

[n le 2 select 2*n+1 else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

Mathematica

RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == 4 a[n - 1] + 2 a[n - 2]}, a, {n, 23}]
LinearRecurrence[{4, 2}, {3, 5}, 24]
Table[((18 + Sqrt[6]) (2 - Sqrt[6])^n - (Sqrt[6] - 18) (2 + Sqrt[6])^n)/12, {n, 0, 23}]

PARI

Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

Formula

G.f.: (3 - 7*x)/(1 - 4*x - 2*x^2).

a(n) = ((18 + sqrt(6))*(2 - sqrt(6))^n - (sqrt(6) - 18)*(2 + sqrt(6))^n)/12.

Lim_{n -> infinity} a(n + 1)/a(n) = 2 + sqrt(6) = A176213.

a(n) = 3*A090017(n+1) -7*A090017(n). - R. J. Mathar, Mar 12 2017