A158920 Binomial transform of A008805 (triangular numbers with repeats).

1, 2, 6, 16, 41, 102, 248, 592, 1392, 3232, 7424, 16896, 38144, 85504, 190464, 421888, 929792, 2039808, 4456448, 9699328, 21037056, 45481984, 98041856, 210763776, 451936256, 966787072, 2063597568, 4395630592, 9344909312, 19830669312

Offset

1,2

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Formula

A007318 * (1, 1, 3, 3, 6, 6, 10, 10, 15, 15, ...) = binomial transform of triangular numbers A000217 with repeats.

From R. J. Mathar, Apr 02 2009: (Start)

G.f.: x*(x-1)^4/(1-2*x)^3.

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 5. (End)

32*a(n) = 2^(n+1) + 3*A001787(n+1) + A001788(n+1), n>=3. - R. J. Mathar, Feb 25 2023

Example

a(4) = 16 = (1, 3, 3, 1) dot (1, 1, 3, 3) = (1 + 3 + 9 + 3).

Maple

A000217 := proc(n) n*(n+1)/2 ; end: A008805 := proc(n) A000217( 1+floor(n/2) ) ; end: L := [seq(A008805(n), n=0..100)] ; read("transforms"); BINOMIAL(L) ; # R. J. Mathar, Apr 02 2009

Mathematica

Join[{1, 2}, LinearRecurrence[{6, -12, 8}, {6, 16, 41}, 30]] (* Harvey P. Dale, Feb 25 2012 *)

Crossrefs

Cf. A008805, A045891.

Sequence in context: A074405 A068786 A276359 * A263592 A178438 A365548

Adjacent sequences: A158917 A158918 A158919 * A158921 A158922 A158923

Keyword

nonn,easy

Author

Gary W. Adamson, Mar 30 2009

Status

approved