A083878 a(0)=1, a(1)=3, a(n) = 6*a(n-1) - 7*a(n-2), n >= 2.

1, 3, 11, 45, 193, 843, 3707, 16341, 72097, 318195, 1404491, 6199581, 27366049, 120799227, 533233019, 2353803525, 10390190017, 45864515427, 202455762443, 893682966669, 3944907462913, 17413664010795, 76867631824379

Offset: 0

Author

Paul Barry, May 08 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A006012. Second binomial transform of A001333.

Third binomial transform of A077957. Inverse binomial transform of A083879. - Philippe Deléham, Dec 01 2008

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1551
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A083879, A056241, A081179, A098158.

Programs

Mathematica

f[n_] := Simplify[(3 + Sqrt@2)^n + (3 - Sqrt@2)^n]/2; Array[f, 23, 0] (* Robert G. Wilson v, Oct 31 2010 *)

Formula

a(n) = ((3 - sqrt(2))^n + (3 + sqrt(2))^n)/2;

a(n) = Sum_{k=0..n} C(n, 2k)*3^(n-2k)*2^k;

G.f.: (1-3x)/(1-6x+7x^2);

E.g.f.: exp(3x)*cosh(x*sqrt(2)).

a(n) = Sum_{k=0..n} C(n, k)*2^((n-k)/2)(1+(-1)^(n-k))*3^k/2. - Paul Barry, Jan 22 2005

a(n) = Sum_{k=0..n} A098158(n,k)*3^(2k-n)*2^(n-k). - Philippe Deléham, Dec 01 2008

a(n) = A081179(n+1) - 3*A081179(n). - R. J. Mathar, Nov 10 2013

a(n) = Sum_{k=1..n} A056241(n, k) * 2^(k-1). - J. Conrad, Nov 23 2022