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
Examples
a(4) = 16 = (1, 3, 3, 1) dot (1, 1, 3, 3) = (1 + 3 + 9 + 3).
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).
Programs
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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
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Mathematica
Join[{1,2},LinearRecurrence[{6,-12,8},{6,16,41},30]] (* Harvey P. Dale, Feb 25 2012 *)
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)