A133654 a(n) = 2*A000129(n) - 1.

1, 3, 9, 23, 57, 139, 337, 815, 1969, 4755, 11481, 27719, 66921, 161563, 390049, 941663, 2273377, 5488419, 13250217, 31988855, 77227929, 186444715, 450117361, 1086679439, 2623476241, 6333631923, 15290740089, 36915112103, 89120964297, 215157040699

Offset: 1

Author

Gary W. Adamson, Sep 19 2007

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to (1 + sqrt(2)).

Define a triangle by T(n,1) = n*(n-1)+1 and T(n,n) = 1, n >= 1. Let interior terms be T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The triangle is 1; 3,1; 7,5,1; 13,15,7,1; etc. The row sums are 1, 4, 13, 36, 93, ... and the differences (sum of terms in row(n) minus those in row(n-1)) are a(n). - J. M. Bergot, Mar 10 2013

Examples

			a(3) = 2*A000129(3) - 1 = 2*5 - 1.
a(5) = 57 = 2*a(4) + a(3) + 2 = 2*23 + 9 + 2.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Crossrefs

Cf. A000129.

Programs

PARI

Vec(x*(1+x^2)/((x-1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, Mar 16 2016

Formula

a(n) = 2*A000129(n) - 1, where A000129 = the Pell sequence. a(1) = 1, a(2) = 3, then for n>2, a(n) = 2*a(n-1) + a(n-2) + 2.

G.f.: x*(1+x^2)/( (x-1)*(x^2+2*x-1) ). - R. J. Mathar, Nov 14 2007

a(n) = -1+(-(1-sqrt(2))^n+(1+sqrt(2))^n)/sqrt(2). - Colin Barker, Mar 16 2016

Extensions

More terms from Philippe Deléham, Oct 16 2007, corrected by R. J. Mathar, Mar 12 2013