A109795 a(n)= n*(1+floor(n/10)) has arbitrary "10" and can generalize

[Peter Pein]

Seemingly the generating functions are (in Mathematica-syntax):

gf[k_] := (Sum[x^i, {i, 1, k - 1}] + (k + 1) x^k) /
          ((1 - x)^3*Sum[x^i, {i, 0, k - 1}]^2)

Cheers,
Peter

P.S.: If you want to check it, here is the whole content of the Mma-notebook:

In[1]:= gf[k_] := Sum[n*(1 + Floor[n/k])*x^n, {n, 0, Infinity}]
In[2]:= gf1to10 = Together /@ gf /@ Range[10]
Out[2]= {-((2*x)/(-1 + x)^3), (-x - 3*x^2)/((-1 + x)^3*(1 + x)^2),
  (-x - x^2 - 4*x^3)/((-1 + x)^3*(1 + x + x^2)^2),
  (-x - x^2 - x^3 - 5*x^4)/((-1 + x)^3*(1 + x + x^2 + x^3)^2),
  (-x - x^2 - x^3 - x^4 - 6*x^5)/((-1 + x)^3*
    (1 + x + x^2 + x^3 + x^4)^2),
  (-x - x^2 - x^3 - x^4 - x^5 - 7*x^6)/
   ((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5)^2),
  (-x - x^2 - x^3 - x^4 - x^5 - x^6 - 8*x^7)/
   ((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^2),
  (-x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - 9*x^8)/
   ((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^2),
  (-x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - 10*x^9)/
   ((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 +
      x^8)^2), (-x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 -
    x^9 - 11*x^10)/((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 +
      x^6 + x^7 + x^8 + x^9)^2)}
In[3]:= gfk[k_] := (Sum[x^i, {i, 1, k - 1}] + (k + 1)*x^k)/
   ((1 - x)^3*Sum[x^i, {i, 0, k - 1}]^2)
In[4]:= Simplify[gf[20] - gfk[20]]
Out[4]= 0

Jonathan Post schrieb:
> Replace the arbitrary "10" in A109795 with the integer k>1 to get the
> array A[k,n] = n*(1+floor(n/k))
> which begins:
> 
> ......|n=1.|.n=2.|.n=3.|.n=4.|.n=5.|.n=6.|.n=7.|.n=8.|.n=9.|.n=10.|
> k=2.|...1.|.....4.|....6.|...12.|....15.|...24.|...28.|....40.|..45.|....60.|
> k=3.|...1.|.....2.|....6.|....8.|.....10.|...18.|...21.|....24.|..36.|....40.|
> k=4.|...1.|.....2.|....3.|....8.|.....10.|...12.|...14.|....24.|..27.|....30.|
> k=5.|...1.|.....2.|....3.|....4.|.....10.|...12.|...14.|....16.|..18.|....30.|
> k=6.|...1.|.....2.|....3.|....4.|.......5.|...12.|...14.|....16.|..18.|....20.|
> k=7.|...1.|.....2.|....3.|....4.|.......5.|.....6.|...14.|....16.|..18.|....20.|
> k=8.|...1.|.....2.|....3.|....4.|.......5.|.....6.|.....7.|....16.|..18.|....20.|
> k=9.|...1.|.....2.|....3.|....4.|.......5.|.....6.|.....7.|......8.|..18.|....20.|
> k=10|..1.|.....2.|....3.|....4.|.......5.|.....6.|.....7.|......8.|....9.|....20.|.A109795
> 
> he rows other than k=10 seem not to be in OEIS.
> 
> The array, by antidiagonals, is:
> 1, 1, 4, 1, 2, 6, 1, 2, 6, 12, 1, 2, 3, 8, 15, 1, 2, 3, 8, 10, 24, 1,
> 2, 3, 4, 10, 18, 28, 1, 2, 3, 4, 10, 12, 21, 40, ...
>