A083878 a(0)=1, a(1)=3, for n >= 2, a(n) = 6*a(n-1) - 7*a(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, 339310142870709, 1497787434453601, 6611553606626643, 29184809598584651

Offset

0,2

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).

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

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

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

Mathematica

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

Prog

(Magma)
[n le 2 select 3^(n-1) else 6*Self(n-1) -7*Self(n-2): n in [1..40]]; // G. C. Greubel, Feb 02 2026
(SageMath)
@CachedFunction
def A083878(n):
    if n<2 : return 3^n
    else: return 6*A083878(n-1) -7*A083878(n-2)
print([A083878(n) for n in range(41)]) # G. C. Greubel, Feb 02 2026

Crossrefs

Cf. A001333, A006012, A056241, A081179, A083879, A098158.

Sequence in context: A238578 A151113 A151114 * A363183 A151115 A151116

Adjacent sequences: A083875 A083876 A083877 * A083879 A083880 A083881

Keyword

easy,nonn

Author

Paul Barry, May 08 2003

Status

approved