A087161 Records in A087159, i.e., A087159(a(n)) = n, and satisfies the recurrence a(n+3) = 5*a(n+2) - 6* a(n+1) + 2*a(n) with a(1) = 1, a(2) = 2, and a(3) = 4.

1, 2, 4, 10, 30, 98, 330, 1122, 3826, 13058, 44578, 152194, 519618, 1774082, 6057090, 20680194, 70606594, 241065986, 823050754, 2810071042, 9594182658, 32756588546, 111837988866, 381838778370, 1303679135746, 4451038986242

Offset: 1

Author

Paul D. Hanna, Aug 22 2003

Keywords

• nonn

Comments

Binomial transform of A001333 (which, with an extra leading 1, is the expansion of (1 - x - 2*x^2)/(1 - 2*x - x^2)). - Paul Barry, Aug 26 2003

Partial sums of the binomial transform of Pell(n-1). - Paul Barry, Apr 24 2004

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Crossrefs

Cf. A000129, A007070, A087159, A087160, A100631.

Programs

Mathematica

CoefficientList[Series[(1-3x)/(1-5x+6x^2-2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{5,-6,2},{1,2,4},30] (* Harvey P. Dale, Oct 12 2015 *)

Formula

G.f.: x*(1 - 3*x)/(1 - 5*x + 6*x^2 - 2*x^3).

a(n) = 2 + 2*A007070(n-3) for n > 2.

a(n) = ((2 - sqrt(2))^(n)/(1 - sqrt(2)) + (2 + sqrt(2))^(n)/(1 + sqrt(2)))/2 + 2 (offset 0) - Paul Barry, Aug 26 2003

a(n+1) - a(n) = A006012(n-1) for n >= 2. - Philippe Deléham, Feb 01 2012

a(1) = 1, a(2) = 2, a(3) = 4, a(n) = 5*a(n-1) - 6*a(n-2) + 2*a(n-3) for n >= 4. - Harvey P. Dale, Oct 12 2015

a(n+1) = Sum_{k=0..n} A100631(n,k) for n >= 0. - Petros Hadjicostas, Feb 09 2021

Extensions

More terms from Paul Barry, Apr 24 2004