1,4

Row sums are: {1, 2, 4, 8, 17, 34, 64, 120, 227, 424, 788, ...}.

Blackmore, D. and Kappraff, J. "Phyllotaxis and Toral Dynamical Systems." ZAMM (1995).

Input vector starting as all ones: b_m[n]=Sum[b_m[n-k],{k,1,m+1}]

{1},

{1, 1},

{2, 1, 1},

{3, 3, 1, 1},

{5, 5, 5, 1, 1},

{8, 9, 10, 5, 1, 1},

{13, 17, 17, 9, 6, 1, 1},

{21, 31, 31, 17, 11, 7, 1, 1},

{34, 57, 59, 33, 21, 13, 8, 1, 1},

{55, 105, 107, 65, 41, 25, 15, 9, 1, 1},

{89, 193, 189, 129, 81, 49, 29, 17, 10, 1, 1}

Clear[a, b, c, d, e, f, g, h, i, j, k]

a[0] := 1; a[1] := 1;

a[n_] := a[n] = a[n - 1] + a[n - 2];

w[1] = Table[a[n], {n, 0, 10}];

b[0] = 1; b[1] = 1; b[2] = 1;

b[n_] := b[n] = b[n - 1] + b[n - 2] + b[n - 3];

w[2] = Table[b[n], {n, 0, 10}];

c[0] = 1; c[1] = 1; c[2] = 1; c[3] = 1;

c[n_] := c[n] = c[n - 1] + a[n - 2] + c[n - 3] + c[n - 4];

w[3] = Table[c[n], {n, 0, 10}];

d[0] = 1; d[1] = 1; d[2] = 1; d[3] = 1; d[4] = 1;

d[n_] := d[n] = d[n - 1] + d[n - 2] + d[n - 3] + d[n - 4] + d[n - 5];

w[4] = Table[d[n], {n, 0, 10}];

e[0] = 1; e[1] = 1; e[2] = 1; e[3] = 1; e[4] = 1; e[5] = 1;

e[n_] := e[n] = e[n - 1] + e[n - 2] + e[n - 3] + e[n - 4] + e[n - 5] + e[n - 6];

w[5] = Table[e[n], {n, 0, 10}];

f[0] = 1; f[1] = 1; f[2] = 1; f[3] = 1; f[4] = 1; f[5] = 1; f[6] = 1;

f[n_] := f[n] = f[n - 1] + f[n - 2] + f[n - 3] + f[n - 4] + f[n - 5] + f[n - 6] + f[n - 7];

w[6] = Table[f[n], {n, 0, 10}];

g[0] = 1; g[1] = 1; g[2] = 1; g[3] = 1; g[4] = 1; g[5] = 1; g[6] = 1; g[ 7] = 1;

g[n_] := g[n] = g[n - 1] + g[n - 2] + g[n - 3] + g[n - 4] + g[n - 5] + g[n - 6] + g[n - 7] + g[n - 8];

w[7] = Table[g[n], {n, 0, 10}];

h[0] = 1; h[ 1] = 1; h[2] = 1; h[3] = 1; h[4] = 1; h[5] = 1; h[6] = 1; h[7] = 1; h[8] = 1;

h[n_] := h[n] = h[n - 1] + h[n - 2] + h[n - 3] + h[n - 4] + h[n - 5] + h[n - 6] + h[n - 7] + h[n - 8] + h[n - 9];

w[8] = Table[h[n], {n, 0, 10}];

i[0] = 1; i[1] = 1;

i[2] = 1; i[3] =

1; i[4] = 1; i[5] = 1; i[6] = 1; i[7] = 1; i[8] = 1; i[9] = 1;

i[n_] := i[n] = i[n - 1] + i[n - 2] + i[n - 3] + i[n - 4] +

i[n - 5] + i[n - 6] + i[n - 7] + i[n - 8] + i[n - 9] + i[n - 10];

w[9] = Table[i[n], {n, 0, 10}];

j[0] = 1; j[1] = 1; j[2] = 1; j[3] = 1; j[4] = 1; j[5] = 1; j[6] = 1; j[ 7] = 1; j[8] = 1; j[9] = 1; j[10] = 1;

j[n_] := j[n] = j[n - 1] + j[n - 2] + j[n - 3] + j[n - 4] +

j[n - 5] + j[n - 6] + j[n - 7] + j[n - 8] + j[n - 9] + j[n - 10] + j[n - 11];

w[10] = Table[j[n], {n, 0, 10}];

k[0] = 1; k[1] = 1; k[2] = 1; k[3] = 1; k[4] = 1; k[5] = 1; k[6] = 1;

k[7] = 1; k[8] = 1; k[9] = 1; k[10] = 1; k[11] = 1;

k[n_] := k[n] = k[n - 1] + k[n - 2] + k[n - 3] + k[n - 4] + k[n - 5] + k[n - 6] + k[n - 7] + k[n - 8] + k[n - 9] + k[n - 10] + k[n - 11] + k[n - 12];

w[11] = Table[k[n], {n, 0, 10}];

Table[Table[w[m][[n]], {m, 1, n}], {n, 1, 11}]

Flatten[%]

Cf. A000045, etc.

Sequence in context: A167040 A054450 A053538 this_sequence A154221 A026736 A050446

Adjacent sequences: A138198 A138199 A138200 this_sequence A138202 A138203 A138204

nonn,uned,tabl

Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), May 04 2008