A173568 Triangle T(n, k) = f(k, n-k+1) + f(n-k+1, k), where f(n, k) = round( ((1+sqrt(k))^(2*n+1) - (1-sqrt(k))^(2*n+1))/(2*sqrt(k))) - 1, read by rows.

6, 19, 19, 68, 56, 68, 261, 211, 211, 261, 1030, 1044, 654, 1044, 1030, 4103, 5819, 2993, 2993, 5819, 4103, 16392, 33560, 19102, 9840, 19102, 33560, 16392, 65545, 195147, 137571, 52989, 52989, 137571, 195147, 65545, 262154, 1136836, 1019606, 412700, 182270, 412700, 1019606, 1136836, 262154

Offset

1,1

Links

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

Formula

T(n, k) = f(k, n-k+1) + f(n-k+1, k), where f(n, k) = round( ((1+sqrt(k))^(2*n+1) - (1-sqrt(k))^(2*n+1))/(2*sqrt(k)) ) - 1.

Example

Triangle begins as:
       6;
      19,      19;
      68,      56,      68;
     261,     211,     211,    261;
    1030,    1044,     654,   1044,   1030;
    4103,    5819,    2993,   2993,   5819,   4103;
   16392,   33560,   19102,   9840,  19102,  33560,   16392;
   65545,  195147,  137571,  52989,  52989, 137571,  195147,   65545;
  262154, 1136836, 1019606, 412700, 182270, 412700, 1019606, 1136836, 262154;

Mathematica

f[n_, k_]:= Round[((1+Sqrt[k])^(2*n+1) - (1-Sqrt[k])^(2*n+1))/(2*Sqrt[k])] - 1;
T[n_, k_]:= f[k, n-k+1] + f[n-k+1, k];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 26 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, k): return round(((1+sqrt(k))^(2*n+1) -(1-sqrt(k))^(2*n+1))/(2*sqrt(k))) -1
def T(n, k): return f(k, n-k+1) + f(n-k+1, k)
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 26 2021

Crossrefs

Sequence in context: A245869 A370155 A184197 * A012589 A009048 A294313

Adjacent sequences: A173565 A173566 A173567 * A173569 A173570 A173571

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Feb 22 2010

Extensions

Edited by G. C. Greubel, Apr 26 2021

Status

approved