A192245 1-sequence of reduction of triangular number sequence by x^2 -> x+1.

0, 3, 9, 29, 74, 179, 403, 871, 1816, 3686, 7316, 14258, 27362, 51827, 97067, 180027, 331038, 604125, 1095085, 1973095, 3535810, 6305148, 11193384, 19790484, 34860084, 61193859, 107080413, 186826121, 325073906, 564190391

Offset: 1

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn
• changed

Comments

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

Crossrefs

Cf. A192232, A192244, A000217.

Programs

Maple

with(combinat, fibonacci); seq((1/2)*(sum(fibonacci(i)*(i^2+3*i+2), i=0..n-1)), n=1..40) # John M. Campbell, Feb 06 2016

Mathematica

(* See A192244. *)

Formula

Empirical g.f.: x^2*(3-3*x+2*x^2)/(1-x)/(1-x-x^2)^3. - Colin Barker, Feb 10 2012

a(n) = (1/2)*Sum_{i=0..n-1} A000045(i)*(i^2+3*i+2). - John M. Campbell, Feb 06 2016. [I assume this is also an empirical observation, not a theorem? - N. J. A. Sloane, Feb 28 2016]