A247976 Triangle read by rows: T(n,k) generated by m-gon expansions in the case of odd m with "vertex to vertex" version or even m with "vertex to side" version. (See comment for details.)

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7

Offset

1,6

Comments

Refer to triangle expansions in A061777 and A101946 (and their companions for m-gons) which are "vertex to vertex" and "vertex to side" versions respectively. The label values at each iteration can be arranged as triangle. Any m-gon can also be arranged as the same triangle with conditions: (i) m is odd and expansion is "vertex to vertex" version or (ii) m is even and expansion is "vertex to side" version. m*Sum_{i=1..k}T(n,k) gives the total label value in n-th iteration. See illustration.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Kival Ngaokrajang, Illustration of initial terms

Formula

T(n, k) = ( T(n-1, k) if k <= (n+1)/2) otherwise T(n-1, k-1) + T(n-1, k) ) for odd n rows, ( T(n-1, k-1) + T(n-1, k) if k < (n+2)/2 otherwise T(n, k - n/2) ) for even n rows, with T(n, 1) = 1 and T(n, n) = floor((n+1)/2). - G. C. Greubel, Feb 18 2022

Example

Triangle begins:
  1;
  1,  1;
  1,  1,  2;
  1,  2,  1,  2;
  1,  2,  1,  3,  3;
  1,  3,  3,  1,  3,  3;
  1,  3,  3,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  4,  6,  4;
  1,  4,  6,  4,  1,  5, 10, 10,  5;
  1,  5, 10, 10,  5,  1,  5, 10, 10, 5;
  ...

Mathematica

T[n_, k_]:= T[n, k]= If[k==1, 1, If[k==n, Floor[(n+1)/2], If[OddQ[n], If[k<=(n+ 1)/2, T[n-1, k], T[n-1, k-1] + T[n-1, k]], If[k<n/2 +1, T[n-1, k-1] + T[n-1, k], T[n, k-n/2] ]]]];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Feb 18 2022 *)

Prog

(Small Basic)
a[1][1]=1
a[2][1]=1
a[2][2]=1
TextWindow.Write(1+", "+1+", "+1+", ")
for n=3 To 30
  If Math.Remainder(n, 2)=1 then
    '===========odd n=================
    j=1
    for k=1 To n
      If k <= (n/2+1/2) then
         a[n][k]=a[n-1][k]
       Else
         a[n][k]=a[n][j]+a[n][j+1]
         j=j+1
       EndIf
       TextWindow.Write(a[n][k]+", ")
     EndFor
  else
    '==============even n===============
    For k=1 To n
      If k <=1 Then
        a[n][k]=1
      Else
        If k < n/2+1 then
          a[n][k]=a[n-1][k-1]+a[n-1][k]
        Else
          a[n][k]=a[n][k-n/2]
        EndIf
      EndIf
      TextWindow.Write(a[n][k]+", ")
    EndFor
  EndIf
EndFor
(Sage)
@CachedFunction
def T(n, k): # A247976
    if (k==1): return 1
    elif (k==n): return (n+1)//2
    elif (n%2==1): return T(n-1, k) if (k <= (n+1)/2) else T(n-1, k-1) + T(n-1, k)
    else: return T(n-1, k-1)+T(n-1, k) if (k < (n+2)/2) else T(n, k-n/2)
flatten([[T(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Feb 18 2022

Crossrefs

Rows sum: A027383.

Column (start from 1s): c3=A008805, c4=A058187, c5=A000332 repeated, c6=A000389 repeated, c7=A000579 repeated.

Vertex to vertex: A061777, A247618, A247619, A247620.

Vertex to side: A101946, A247903, A247904, A247905.

Cf. A074909.

Sequence in context: A161261 A161286 A185216 * A280986 A344773 A103689

Adjacent sequences: A247973 A247974 A247975 * A247977 A247978 A247979

Keyword

tabl,nonn

Author

Kival Ngaokrajang, Sep 28 2014

Status

approved