A119997 Sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j)*Fibonacci[i+j-1].

1, 1, 4, 5, 17, 32, 97, 225, 628, 1573, 4225, 10880, 28769, 74849, 196708, 513765, 1347025, 3523360, 9229441, 24154625, 63251156, 165571781, 433507969, 1134881280, 2971250497, 7778684737, 20365103812, 53316141125, 139584105233, 365434903328, 956722661665

Offset: 1

Author

Alexander Adamchuk, Aug 03 2006

Keywords

• nonn
• easy

Comments

Prime p divides a(p-1) for p={5,11,19,29,31,41,59,61,71,...} = A038872[n] Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. p^2 divides a(p-1) for prime p={11,19,29,31,41,59,61,71,...} = A045468[n] Primes congruent to {1, 4} mod 5. Square prime divisors of a(n) up to n=50 are{2,3,5,7,11,13,19,23,29,31,41,47,89,101,139,151,199,211,461,521,3571,9349}, It appears that square prime divisors of a(n) belong to A061446[n] Primitive part of Fibonacci(n), A001578[n] Smallest primitive prime factor of Fibonacci number F(n) and A072183[n] Sequence arising from factorization of the Fibonacci numbers. Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}] = A120297[n]. Sum[Sum[i+j-1,{i,1,n}],{j,1,n}] = n^3. Sum[Sum[(-1)^(i+j)*(i+j-1),{i,1,n}],{j,1,n}] = n for odd n and = 0 for even n.

Examples

			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
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000 [Terms up to n=200 from _Vincenzo Librandi_]
• Index entries for linear recurrences with constant coefficients, signature (3,1,-7,5,-1).

Crossrefs

Cf. A120297, A000045, A038872, A001924, A062381, A038872, A045468, A061446, A001578, A072183.

Programs

Mathematica

Table[Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

PARI

a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*fibonacci(i+j-1))) \\ Colin Barker, Mar 26 2015

PARI

Vec(-x*(x^3+2*x-1)/((x-1)*(x^2-3*x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Mar 26 2015

Formula

a(n) = Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}].

a(n) = 3*a(n-1)+a(n-2)-7*a(n-3)+5*a(n-4)-a(n-5) for n>5. - Colin Barker, Mar 26 2015

G.f.: -x*(x^3+2*x-1) / ((x-1)*(x^2-3*x+1)*(x^2-x-1)). - Colin Barker, Mar 26 2015