A108851 a(n) = 4*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 2.

1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, 10878914, 50540729, 234799658, 1090820819, 5067682250, 23543191457, 109375812578, 508132824683, 2360658736466, 10967033419913, 50950109889050

Offset: 0

Author

Philippe Deléham, Jul 11 2005

Keywords

• easy
• nonn

Comments

Binomial transform of A083098, second binomial transform of (1, 0, 7, 0, 49, 0, 243, 0, ...).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,3).

Crossrefs

Cf. A080042. - Zerinvary Lajos, May 14 2009

Cf. A015530, A083098.

Appears in A179596, A179597 and A126473. - Johannes W. Meijer, Aug 01 2010

Programs

Magma

[Floor(((2 + Sqrt(7))^n + (2 - Sqrt(7))^n) / 2): n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

Mathematica

LinearRecurrence[{4,3},{1,2},30] (* Harvey P. Dale, Jan 02 2022 *)

PARI

a(n)=round(((2+sqrt(7))^n+(2-sqrt(7))^n)/2) \\ Charles R Greathouse IV, Dec 06 2011

Sage

[lucas_number2(n,4,-3)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = ((2 + sqrt(7))^n + (2 - sqrt(7))^n) / 2.

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

E.g.f.: exp(2*x)*cosh(sqrt(7)*x).

a(n+1)/a(n) converges to 2 + sqrt(7) = 4.645751311064...

Limit_{k->oo} a(n+k)/a(k) = A108851(n) + A015530(n)*sqrt(7); also lim_{n->oo} A108851(n)/A015530(n) = sqrt(7). - Johannes W. Meijer, Aug 01 2010

a(n) = Sum_{k=0..n} A201730(n,k)*6^k. - Philippe Deléham, Dec 06 2011

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013

a(n) = (2 + sqrt(7))^n - A015530(n)*sqrt(7). - Robert FERREOL, Aug 04 2025