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A213587 Rectangular array:  (row n) = b**c, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution. 7
1, 4, 2, 10, 7, 3, 22, 17, 11, 5, 45, 37, 27, 18, 8, 88, 75, 59, 44, 29, 13, 167, 146, 120, 96, 71, 47, 21, 310, 276, 234, 195, 155, 115, 76, 34, 566, 511, 443, 380, 315, 251, 186, 123, 55, 1020, 931, 821, 719, 614, 510, 406, 301, 199, 89, 1819, 1675 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A213588.

Antidiagonal sums: A213589.

Row 1,  (1,2,3,5,...)**(1,2,3,5,...): A004798.

Row 2,  (1,2,3,5,...)**(2,3,5,8,...)

Row 3,  (1,2,3,5,...)**(3,5,8,13,...)

For a guide to related arrays, see A213500.

LINKS

Clark Kimberling, Antidiagonals n = 1..60, flattened

FORMULA

Rows:  T(n,k) = 2*T(n,k-1)+T(n,k-2)-2*T(n,k-3)-T(n,k-4).

Columns:  T(n,k)=T(n-1,k) + T(n-2,k).

G.f. for row n:  f(x)/g(x), where f(x) = F(n+1) + F(n+2)*x + F(n)*x^2 and g(x) = (1 - x - x^2)^2.

EXAMPLE

Northwest corner (the array is read by falling antidiagonals):

1....4....10....22....45....88....167

2....7....17....37....75....146...276

3....11...27....59....120...234...443

5....18...44....96....195...380...719

8....29...71....155...315...614...1162

12...47...115...251...510...994...1881

MATHEMATICA

b[n_] := Fibonacci[n + 1]; c[n_] := Fibonacci[n + 1];

t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]

TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]

Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]

r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213587 *)

Table[t[n, n], {n, 1, 40}] (* A213588 *)

s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]

Table[s[n], {n, 1, 50}] (* A213589 *)

CROSSREFS

Cf. A213500.

Sequence in context: A160572 A213500 A213584 * A066579 A117821 A185732

Adjacent sequences:  A213584 A213585 A213586 * A213588 A213589 A213590

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 19 2012

STATUS

approved

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Last modified November 23 03:36 EST 2014. Contains 249837 sequences.