A172970 Triangle T(n, k) = A172452(n) - A172452(k) - A172452(n-k), read by rows.

-1, -1, -1, -1, -1, -1, -1, 0, 0, -1, -1, 1, 2, 1, -1, -1, 7, 9, 9, 7, -1, -1, 35, 43, 44, 43, 35, -1, -1, 143, 179, 186, 186, 179, 143, -1, -1, 575, 719, 754, 760, 754, 719, 575, -1, -1, 3071, 3647, 3790, 3824, 3824, 3790, 3647, 3071, -1, -1, 19199, 22271, 22846, 22988, 23016, 22988, 22846, 22271, 19199, -1

Offset

0,13

Comments

Row sums are: {-1, -2, -3, -2, 2, 30, 198, 1014, 4854, 28662, 197622, ...}.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = A172452(n) - A172452(k) - A172452(n-k).

T(n, k) = c(n) - c(k) - c(n-k) where c(n) = Product_{j=1..n} A004001(j).

Example

Triangle begins as:
  -1;
  -1,    -1;
  -1,    -1,    -1;
  -1,     0,     0,    -1;
  -1,     1,     2,     1,    -1;
  -1,     7,     9,     9,     7,    -1;
  -1,    35,    43,    44,    43,    35,    -1;
  -1,   143,   179,   186,   186,   179,   143,    -1;
  -1,   575,   719,   754,   760,   754,   719,   575,    -1;
  -1,  3071,  3647,  3790,  3824,  3824,  3790,  3647,  3071,    -1;
  -1, 19199, 22271, 22846, 22988, 23016, 22988, 22846, 22271, 19199, -1;

Mathematica

f[n_]:= f[n]= If[n<3, Fibonacci[n], f[f[n-1]] + f[n-f[n-1]]]; (* f=A004001 *)
c[n_]:= Product[f[j], {j, n}]; (* c=A172452 *)
T[n_, k_]:= c[n] - c[k] - c[n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)

Prog

(Sage)
@CachedFunction
def b(n): return fibonacci(n) if (n<3) else b(b(n-1)) + b(n-b(n-1)) # b=A004001
def c(n): return product(b(j) for j in (1..n)) # c=A172452
def T(n, k): return c(n) - c(k) - c(n-k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 27 2021

Crossrefs

Cf. A004001, A172452, A172453.

Sequence in context: A294105 A101124 A011127 * A172971 A306702 A240581

Adjacent sequences: A172967 A172968 A172969 * A172971 A172972 A172973

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 06 2010

Extensions

Edited by G. C. Greubel, Apr 27 2021

Status

approved