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.

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, 1497, 1332, 1162, 994, 825, 657, 487, 322, 144

Offset: 1

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• tabl
• easy

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.

Examples

			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
  13...47...115...251...510...994...1881
		

Links

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

Crossrefs

Cf. A213500, A213588, A213589, A004798.

Programs

GAP

Flat( List([1..12], n-> List([1..n], k-> ((n-k+1)*Lucas(1,-1, n+3)[2] - Fibonacci(n-k+1)*Lucas(1,-1,k-1)[2])/5 ))); # G. C. Greubel, Jul 08 2019

Magma

[[((n-k+1)*Lucas(n+3) - Fibonacci(n-k+1)*Lucas(k-1))/5: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

Mathematica

(* First program *)
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}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
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 *)
(* Second program *)
Table[((n-k+1)*LucasL[n+3] - Fibonacci[n-k+1]*LucasL[k-1])/5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

PARI

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
t(n,k) = ((n-k+1)*lucas(n+3) - fibonacci(n-k+1)*lucas(k-1))/5;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

Sage

[[((n-k+1)*lucas_number2(n+3,1,-1) - fibonacci(n-k+1)* lucas_number2(k-1, 1,-1))/5 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

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.

T(n, k) = (k*Lucas(n+k+2) - Fibonacci(k)*Lucas(n-1))/5. - G. C. Greubel, Jul 08 2019