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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A119986 A245869 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

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Last modified October 6 02:32 EDT 2022. Contains 357261 sequences. (Running on oeis4.)