A213576 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and ** = convolution.

1, 3, 1, 7, 4, 2, 14, 10, 7, 3, 26, 21, 17, 11, 5, 46, 40, 35, 27, 18, 8, 79, 72, 66, 56, 44, 29, 13, 133, 125, 118, 106, 91, 71, 47, 21, 221, 212, 204, 190, 172, 147, 115, 76, 34, 364, 354, 345, 329, 308, 278, 238, 186, 123, 55, 596, 585, 575, 557, 533, 498, 450, 385, 301, 199, 89

Offset

1,2

Comments

Principal diagonal: A213577.

Antidiagonal sums: A213578.

Row 1, (1,2,3,...)**(1,1,2,3,5,...): A001924;

Row 2, (1,2,3,...)**(1,2,3,5,8,...): A001891;

Row 3, (1,2,3,...)**(2,3,5,8,13,...): A033937;

Row 4, (1,2,3,...)**(3,5,8,13,21,...): A033960;

Row 5, (1,2,3,...)**(5,8,13,21,...): A037140;

Row 6, (1,2,3,...)**(8,13,21,34,...): A037157.

For a guide to related arrays, see A213500.

The falling antidiagonal rows can be computed by the sum Sum_{j=0..n-k} (n-k-j+1)*Fibonacci(k+j) which can also be seen as Fibonacci(n+4) - Lucas(k+2) - (n-k)*Fibonacci(k+1). - G. C. Greubel, Jul 05 2019

Links

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

Formula

Rows: T(n,k) = 3*T(n,k-1) - 2*T(n,k-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) - F(n-1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.

T(n,k) = F(n+k+3) - k*F(n+1) - F(n+3). - Ehren Metcalfe, Jul 04 2019

Example

Northwest corner (the array is read by falling antidiagonals):
  1,   3,   7,  14,  26,  46,  79
  1,   4,  10,  21,  40,  72, 125
  2,   7,  17,  35,  66, 118, 204
  3,  11,  27,  56, 106, 190, 329
  5,  18,  44,  91, 172, 308, 533
  8,  29,  71, 147, 278, 498, 862

Mathematica

(* First Program *)
b[n_]:= n; c[n_]:= Fibonacci[n];
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}]] (* A213576 *)
r[n_]:= Table[t[n, k], {k, 1, 40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n, 1, 40}] (* A213577 *)
s[n_]:= Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* Second Program *)
T[n_, k_]:= Fibonacci[n+4] - (n-k)*Fibonacci[k+1] - LucasL[k+2];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

Prog

(PARI) T(n, k)= fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 05 2019
(Magma) [[Fibonacci(n+4) -(n-k)*Fibonacci(k+1) -Lucas(k+2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Jul 05 2019
(Sage) [[fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Jul 05 2019
(GAP) Flat(List([1..10], n-> List([1..n], k-> Fibonacci(n+4) - (n-k+1) *Fibonacci(k+1) - Fibonacci(k+3)))) # G. C. Greubel, Jul 05 2019

Crossrefs

Cf. A213500.

Sequence in context: A193970 A274510 A158841 * A021319 A190177 A283764

Adjacent sequences: A213573 A213574 A213575 * A213577 A213578 A213579

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jun 18 2012

Status

approved