A119997 - OEIS

%I #17 Jun 13 2015 00:52:06

%S 1,1,4,5,17,32,97,225,628,1573,4225,10880,28769,74849,196708,513765,

%T 1347025,3523360,9229441,24154625,63251156,165571781,433507969,

%U 1134881280,2971250497,7778684737,20365103812,53316141125,139584105233,365434903328,956722661665

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

%C 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.

%H Colin Barker, <a href="/A119997/b119997.txt">Table of n, a(n) for n = 1..1000</a> [Terms up to n=200 from _Vincenzo Librandi_]

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-7,5,-1).

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

%F 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

%F G.f.: -x*(x^3+2*x-1) / ((x-1)*(x^2-3*x+1)*(x^2-x-1)). - _Colin Barker_, Mar 26 2015

%e Matrix begins:

%e 1 -1 2 -3 5

%e -1 2 -3 5 -8

%e 2 -3 5 -8 13

%e -3 5 -8 13 -21

%e 5 -8 13 -21 34

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

%o (PARI) a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*fibonacci(i+j-1))) \\ _Colin Barker_, Mar 26 2015

%o (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

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

%K nonn,easy

%O 1,3

%A _Alexander Adamchuk_, Aug 03 2006