A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

1, 4, 3, 10, 8, 5, 21, 18, 12, 7, 40, 35, 26, 16, 9, 72, 64, 49, 34, 20, 11, 125, 112, 88, 63, 42, 24, 13, 212, 191, 152, 112, 77, 50, 28, 15, 354, 320, 257, 192, 136, 91, 58, 32, 17, 585, 530, 428, 323, 232, 160, 105, 66, 36, 19

Offset: 1

Author

Clark Kimberling, Jun 21 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213769.

Antidiagonal sums: A213770.

Row 1, (1,1,2,3,5,...)**(1,3,5,7,9,...): A001891.

Row 2, (1,1,2,3,5,...)**(3,5,7,9,11,...).

Row 3, (1,1,2,3,5,...)**(5,7,9,11,13,...).

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....4....10...21...40....72....125
3....8....18...35...64....112...191
5....12...26...49...88....152...257
7....16...34...63...112...192...323
9....20...42...77...136...232...389
11...24...50...91...160...272...455
		

Links

• Clark Kimberling, Antidiagonals n=1..80, flattened

Crossrefs

Cf. A001891, A213500, A213769, A213770.

Programs

Mathematica

b[n_] := Fibonacci[n]; c[n_] := 2 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}]  (* A213768 *)
Table[t[n, n], {n, 1, 40}] (* A213769 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213770 *)

Formula

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

G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - x - x^2)(1 - x )^2.

T(n,k) = 2*n*Fibonacci(k+2) + Lucas(k+2) - 2*(k+n) - 3. - Ehren Metcalfe, Jul 08 2019