A172970 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

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