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

1,2

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

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

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

Example

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

Prog

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

Crossrefs

Cf. A000129.

Sequence in context: A341029 A045883 A292329 * A193695 A226845 A096574

Adjacent sequences: A133651 A133652 A133653 * A133655 A133656 A133657

Keyword

nonn,easy

Author

Gary W. Adamson, Sep 19 2007

Extensions

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

Status

approved