A120297 Sum of all matrix elements of n X n matrix M(i,j) = Fibonacci(i+j-1).

1, 5, 20, 65, 193, 544, 1489, 4005, 10660, 28193, 74273, 195200, 512257, 1343077, 3519412, 9219105, 24144289, 63224096, 165544721, 433437125, 1134810436, 2971065025, 7778499265, 20364618240, 53315655553, 139582833989

Offset

1,2

Comments

p^2 divides a(p-1) for p = 5, 11, 19, 29, 31, 41, 59, 61, 71, ... = A038872 (Primes congruent to {0, 1, 4} mod 5), also odd primes p such that where 5 is a square mod p. All squared prime divisors of a(n) also belong to A038872.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

a(n) = Sum_{j=1..n} Sum_{i=1..n} Fibonacci(i+j-1).

a(n) = Fibonacci(2*n+3) - 2*Fibonacci(n+3) + 2. - Vladeta Jovovic, Jul 21 2006

G.f.: (1 - x^3 + 2*x^2)/((x-1)*(x^2 + x - 1)*(x^2 - 3*x + 1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009

Example

Matrix begins:
  1  1  2  3  5
  1  2  3  5  8
  2  3  5  8 13
  3  5  8 13 21
  5  8 13 21 34

Mathematica

Table[Sum[Sum[Fibonacci[i+j-1], {i, 1, n}], {j, 1, n}], {n, 1, 50}]

Crossrefs

Cf. A000045, A038872, A001924, A062381.

Sequence in context: A160528 A023004 A001873 * A271066 A271599 A032286

Adjacent sequences: A120294 A120295 A120296 * A120298 A120299 A120300

Keyword

nonn

Author

Alexander Adamchuk, Jul 11 2006

Status

approved