A080960 Third binomial transform of A010685 (period 2: repeat 1,4).

1, 7, 34, 148, 616, 2512, 10144, 40768, 163456, 654592, 2619904, 10482688, 41936896, 167759872, 671064064, 2684305408, 10737319936, 42949476352, 171798298624, 687193980928, 2748777496576, 10995113132032, 43980458819584

Offset: 0

Author

Paul Barry, Mar 03 2003

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (6,-8).

Crossrefs

Cf. A003945, A010685, A048473, A080961.

Programs

Magma

binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >; binomtf(binomtf(binomtf(&cat[ [1, 4]: n in [1..12] ]))); // Klaus Brockhaus, Nov 26 2009

Mathematica

RecurrenceTable[{a[0]==1,a[n]==4a[n-1]+3*2^(n-1)},a,{n,30}] (* or *) LinearRecurrence[{6,-8},{1,7},30] (* Harvey P. Dale, Nov 12 2012 *)
CoefficientList[Series[(1+x)/((1-2x)(1-4x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

Sage

[2^(n-1)*(5*2^n -3) for n in (0..30)] # G. C. Greubel, Nov 23 2021

Formula

a(n) = 4*a(n-1) + 3*2^(n-1).

a(n) = (5/2)*4^n - (3/2)*2^n.

G.f.: (1+x)/((1-2*x)*(1-4*x)). - Klaus Brockhaus, Nov 26 2009

a(n) = 6*a(n-1) - 8*a(n-2), a(0)=1, a(1)=7. - Harvey P. Dale, Nov 12 2012

E.g.f.: exp(2*x)*(5*exp(2*x) - 3)/2. - G. C. Greubel, Nov 23 2021

Extensions

Definition corrected, edited by Klaus Brockhaus, Nov 26 2009