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A059434 Triangle formed when cumulative boustrophedon transform is applied to 1, 1, 1, 1, ..., read by rows in natural order. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified July 28 13:15 EDT 2021. Contains 346332 sequences. (Running on oeis4.)