A059434 Triangle formed when cumulative boustrophedon transform is applied to 1, 1, 1, 1, ..., read by rows in natural order.

1, 1, 2, 1, 3, 6, 1, 7, 16, 26, 1, 27, 69, 118, 168, 1, 169, 455, 810, 1192, 1575, 1, 1576, 4343, 7920, 11952, 16153, 20355, 1, 20356, 56864, 105324, 161704, 222427, 284726, 347026, 1, 347027, 978779, 1832958, 2848841, 3970048, 5148119, 6346546, 7544974

Offset

0,3

Links

Table of n, a(n) for n=0..44.

Formula

From Petros Hadjicostas, Feb 16 2021: (Start)

T(i,j) = T(i,j-1) + Sum_{r=1..j} T(i-1,i-r) for i >= 1 and 1 <= j <= i with T(i,0) = b(i+1) for i >= 0, where b(i) = 1 for i >= 1. (The sequence b = (b(i): i >= 1) is the input sequence.)

T(i,j) = 2*T(i,j-1) - T(i,j-2) + T(i-1,i-j) for i >= 2 and 2 <= j <= i.

T(i,i) = A059430(i) = T(i+1,1) - 1 for i >= 0. (End)

Example

Triangle T(i,j) (with rows i >= 0 and columns j = 0..i) begins:
  1;
  1,    2;
  1,    3,    6;
  1,    7,   16,   26;
  1,   27,   69,  118,   168;
  1,  169,  455,  810,  1192,  1575;
  1, 1576, 4343, 7920, 11952, 16153, 20355;
  ... - Petros Hadjicostas, Feb 16 2021

Maple

# This is a modification of N. J. A. Sloane's program from A059429:
CBOUS2 := proc(a) local c, i, j, n, r: option remember: if whattype(a) <> list then RETURN([]): end if: n := min(nops(a), 60): for i from 0 to n - 1 do c[i, 0] := a[i + 1]: end do: for i to n - 1 do for j to i do c[i, j] := c[i, j - 1] + add(c[i - 1, i - r], r = 1 .. j): end do: end do: RETURN([seq(seq(c[i, j], j = 0 .. i), i = 0 .. n - 1)]): end proc:
# To get the flattened triangle up to the 9th row, we type
CBOUS2([1, 1, 1, 1, 1, 1, 1, 1, 1, 1]); # Petros Hadjicostas, Feb 16 2021

Mathematica

nmax = 9; Clear[CBOUS2, c]; CBOUS2[a_List] := CBOUS2[a] = Module[{i, j, n, r }, n = Min[Length[a], nmax]; For[i = 0, i <= n - 1, i++, c[i, 0] = a[[i + 1]]]; For[i = n - 1, i <= nmax, i++, For[j = 1, j <= i, j++, c[i, j] = c[i, j - 1] + Sum[c[i - 1, i - r], {r, 1, j}]]]; Return[Table[c[i, i], {i, 0, n - 1}]]]; Do[CBOUS2[Table[1, {n}]], {n, 0, nmax}]; Table[c[i, j], {i, 0, nmax - 1}, {j, 0, i}] // Flatten (* Jean-François Alcover, Jul 14 2017, adapted from Maple code for A059430 *)

Crossrefs

Cf. A059429, A059430, A059433.

Sequence in context: A156914 A289656 A248686 * A292222 A182928 A141476

Adjacent sequences: A059431 A059432 A059433 * A059435 A059436 A059437

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jan 31 2001

Extensions

More terms from Floor van Lamoen, Oct 08 2001

Status

approved