A079284 Diagonal sums of triangle A008949.

1, 1, 3, 4, 9, 13, 26, 39, 73, 112, 201, 313, 546, 859, 1469, 2328, 3925, 6253, 10434, 16687, 27633, 44320, 72977, 117297, 192322, 309619, 506037, 815656, 1329885, 2145541, 3491810, 5637351, 9161929, 14799280, 24026745, 38826025, 62983842, 101809867, 165055853, 266865720

Offset: 0

Author

Paul Barry, Feb 08 2003

Keywords

• easy
• nonn

Comments

a(2n) - a(2n-1) = Fibonacci(2n+1).

Diagonal sums of triangle A054450. - Paul Barry, Oct 23 2004

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-2).

Crossrefs

Cf. A000045, A005672, A008949, A061667, A079284, A102517.

Programs

Magma

[Fibonacci(n+3)-2^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 05 2013

Maple

with (combinat):a[0]:=0:a[1]:=1:a[2]:=1:for n from 2 to 50 do a[n]:=fibonacci(n-1)+2*a[n-2] od: seq(a[n], n=1..31); # Zerinvary Lajos, Mar 17 2008

Mathematica

CoefficientList[Series[(1 - x^2) / ((1 - x - x^2) (1 - 2 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{1,3,-2,-2},{1,1,3,4},40] (* Harvey P. Dale, Nov 30 2018 *)

Formula

a(n) = Sum_{j=0..floor(n/2)} Sum_{i=0..j} binomial(n-j, i).

a(n) = Fibonacci(n+3) - 2^floor((n+1)/2). - Vladeta Jovovic, Feb 12 2003

G.f.: (1-x^2)/((1-x-x^2)(1-2x^2)). - Paul Barry, Jan 13 2005