A176481 Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 2, where b(n) = A001333(n).

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 11, 13, 11, 1, 1, 25, 33, 33, 25, 1, 1, 59, 81, 87, 81, 59, 1, 1, 141, 197, 217, 217, 197, 141, 1, 1, 339, 477, 531, 545, 531, 477, 339, 1, 1, 817, 1153, 1289, 1337, 1337, 1289, 1153, 817, 1, 1, 1971, 2785, 3119, 3249, 3283, 3249, 3119, 2785, 1971, 1

Offset

0,5

Comments

Row sums are: {1, 2, 5, 12, 37, 118, 369, 1112, 3241, 9194, 25533, ...}.

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

B. Adamczewski, Ch. Frougny, A. Siegel and W. Steiner, Rational numbers with purely periodic beta-expansion, arXiv:0907.0206 [math.NT], 2009-2010; Bull. Lond. Math. Soc. 42:3 (2010), pp. 538-552.

Formula

Let b(n) = ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 = A001333(n), then T(n, k) = b(n) - b(k) - b(n-k) + 2.

Example

Triangle begins as:
  1;
  1,    1;
  1,    3,    1;
  1,    5,    5,    1;
  1,   11,   13,   11,    1;
  1,   25,   33,   33,   25,    1;
  1,   59,   81,   87,   81,   59,    1;
  1,  141,  197,  217,  217,  197,  141,    1;
  1,  339,  477,  531,  545,  531,  477,  339,    1;
  1,  817, 1153, 1289, 1337, 1337, 1289, 1153,  817,    1;
  1, 1971, 2785, 3119, 3249, 3283, 3249, 3119, 2785, 1971, 1;

Mathematica

b[n_]:= LucasL[n, 2]/2; T[n_, k_]:= b[n] -b[k] -b[n-k] +2;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 06 2019 *)

Prog

(PARI)
{b(n) = round(((1+sqrt(2))^n + (1-sqrt(2))^n)/2)};
{T(n, k) = b(n) -b(k) -b(n-k) +2};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 06 2019
(Magma) b:= func< n| Round(((1+Sqrt(2))^n + (1-Sqrt(2))^n)/2) >; [[b(n)-b(k)-b(n-k)+2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 06 2019
(Sage)
def b(m): return lucas_number2(m, 2, -1)/2
def T(n, k): return b(n) - b(k) - b(n-k) + 2
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 06 2019

Crossrefs

Cf. A001333.

Sequence in context: A056152 A171229 A125690 * A108553 A176700 A300539

Adjacent sequences: A176478 A176479 A176480 * A176482 A176483 A176484

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 18 2010

Extensions

Edited by G. C. Greubel, May 06 2019

Status

approved