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
Kalika Prasad and Munesh Kumari, The generalized k-Leonardo numbers: a non-homogeneous generalization of the Fibonacci numbers, Palestine J. Math. (2025) Vol. 14, No. 3, 637-650. See pp. 638-639.
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.
MATHEMATICA
LinearRecurrence[{3, -1, -1}, {1, 3, 9}, 30] (* Paolo Xausa, Nov 08 2025 *)
PROG
(PARI) Vec(x*(1+x^2)/((x-1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, Mar 16 2016
CROSSREFS
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