A081336 a(n) = (7^n + 3^n)/2.

1, 5, 29, 185, 1241, 8525, 59189, 412865, 2885681, 20186645, 141267149, 988751945, 6920909321, 48445302365, 339113927909, 2373787929425, 16616486808161, 116315321563685, 814206992665469, 5699448173817305, 39896134892198201

Offset: 0

Author

Paul Barry, Mar 18 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A081336.

5th binomial transform of (1,0,4,0,16,0,64,...).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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 (10,-21).

Crossrefs

Cf. A074608, A081337.

Programs

Magma

[(7^n+3^n)/2: n in [0..25]]; // Vincenzo Librandi, Aug 08 2013

Mathematica

CoefficientList[Series[(1 - 5 x) / ((1 - 3 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{10,-21},{1,5},30] (* Harvey P. Dale, Dec 07 2014 *)

PARI

a(n)=(7^n+3^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 10*a(n-1) - 21*a(n-2), a(0)=1, a(1)=5.

G.f.: (1-5*x)/((1-3*x)*(1-7*x)).

E.g.f.: exp(5*x) * cosh(2*x).

a(n) = A074608(n) / 2. - Michel Marcus, Oct 07 2015

a(n) = Sum_{k=0..n} A027907(n,2k)*4^k . - J. Conrad, Aug 24 2016