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A294453 Array read by antidiagonals: T(0,k) = A000045(k+1) for k >= 0. T(n,0) = 1 for n >= 0; thereafter T(n,k) = T(n-1,k-1)+T(n-1,k) for n, k >= 1. 1
1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 5, 5, 5, 1, 5, 8, 8, 8, 8, 1, 6, 12, 13, 13, 13, 13, 1, 7, 17, 21, 21, 21, 21, 21, 1, 8, 23, 33, 34, 34, 34, 34, 34, 1, 9, 30, 50, 55, 55, 55, 55, 55, 55, 1, 10, 38, 73, 88, 89, 89, 89, 89, 89, 89, 1, 11, 47, 103, 138, 144, 144, 144, 144, 144, 144, 144, 1, 12, 57 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

This is another "Fibonacci array".

LINKS

Robert Israel, Table of n, a(n) for n = 0..10010 (antidiagonals 0 to 140, flattened)

Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Table 6.

FORMULA

G.f. as array: 1/((1-x-x*y)*(1-y-y^2)). - Robert Israel, Nov 22 2017

EXAMPLE

The array begins:

1, 1,  2,   3,   5,   8,  13,  21,   34,...

1, 2,  3,   5,   8,  13,  21,  34,   55,...

1, 3,  5,   8,  13,  21,  34,  55,   89,...

1, 4,  8,  13,  21,  34,  55,  89,  144,...

1, 5, 12,  21,  34,  55,  89, 144,  233,...

1, 6, 17,  33,  55,  89, 144, 233,  377,...

1, 7, 23,  50,  88, 144, 233, 377,  610,...

1, 8, 30,  73, 138, 232, 377, 610,  987,...

1, 9, 38, 103, 211, 370, 609, 987, 1597,...

...

The first few antidiagonals are:

1,

1,1,

1,2,2,

1,3,3,3,

1,4,5,5,5,

1,5,8,8,8,8,

1,6,12,13,13,13,13,

1,7,17,21,21,21,21,21,

...

MAPLE

A294453:= proc(n, k) option remember;

  if n = 0 then combinat:-fibonacci(k+1)

  elif k = 0 then 1

  else procname(n-1, k-1)+procname(n-1, k)

  fi

end proc:

seq(seq(A294453(m-k, k), k=0..m), m=0..10); # Robert Israel, Nov 22 2017

MATHEMATICA

T[0, k_] := Fibonacci[k + 1];

T[_, 0] = 1;

T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k];

Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Oct 16 2020 *)

CROSSREFS

Cf. A000045, A001629 (sums of antidiagonals).

Sequence in context: A210798 A117501 A117915 * A097094 A210870 A104726

Adjacent sequences:  A294450 A294451 A294452 * A294454 A294455 A294456

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Nov 22 2017

EXTENSIONS

Better definition and more terms from Robert Israel, Nov 22 2017

STATUS

approved

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Last modified April 19 00:18 EDT 2021. Contains 343098 sequences. (Running on oeis4.)