A072183 -id:A072183 - cp's OEIS frontend

A072270 A partial product representation of A006131 and A072265.

1, 1, 13, 9, 101, 5, 701, 49, 361, 29, 31021, 33, 204101, 181, 1021, 1889, 8799541, 233, 57746701, 1361, 41581, 7589, 2486401661, 1633, 161532401, 49661, 22810681, 58241, 702418373381, 2245, 4608956945501, 3437249, 74991181, 2135149, 2802699901, 75921, 1302034904649701, 14007941, 3219888061, 3019201

Offset: 1

Miklos Kristof, Jul 09 2002

nonn

Define f(n) = A006131(n-1) and L(n) = 4*f(n-1)+f(n+1), which implies L(n) = A072265(n), n>1.
For even n, f(n) = product_{d|n} a(d) and for odd n, f(n) = product_{d|n} a(2d).
Taking logarithms defines the sequence a(.) via a Mobius transformation (see A072183).
Writing f(n) and L(n) in terms of Binet formulas leads to a representation as cyclotomic polynomials.

			f(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 1*1*13*9*5*33 = 19305 for even n=12.
f(9)=a(2)*a(6)*a(18)= 1*5*233 = 1165 for odd n=9.
L(6)=a(4)*a(12) = 9*33 = 297 = 4*f(5)+f(7) = 4*29+181 for even n=6.
L(15)=a(1)*a(3)*a(5)*a(15) = 1*13*101*1021 = 1340573 for odd n=15.

Cf. A006131, A072183.

A072270 := proc(n) if n <=2 then 1; else h := (1+sqrt(17))/2 ; cy := numtheory[cyclotomic](n,x) ; g := degree(cy) ; (h-1)^g*subs(x=h^2/4,cy) ; expand(%) ; end if; end proc: # R. J. Mathar, Nov 17 2010

Let h=(1+sqrt(17))/2, Phi(n, x) = n-th cyclotomic polynomial, so x^n-1= product_{d|n} Phi(d, x), and let g(d) be the order of Phi(d, x). Then a(n)=(h-1)^g(n)*Phi(n, h^2/4), n>2.
a(p) = L(p) for odd prime p.
a(2p) = f(p) for odd prime p.
a(2^k+1) = L(2^k).
a(3*2^k = L(2^k)-4^k.
L(n) = product_{d|n} a(d) for odd n.
L(n*2^k) = product_{d|n} a(d*2^(k+1)) for k>0 and odd n.

Divided argument of Phi by 4; moved comments to formula section - R. J. Mathar, Nov 17 2010

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

Original entry on oeis.org

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

Alexander Adamchuk, Aug 03 2006

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.

			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

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).

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

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

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

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

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

A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

Original entry on oeis.org

1, 2, 2, 7, 2, 1, 11, 3, 11, 1, 29, 47, 29, 1, 19, 41, 19, 1, 199, 23, 199, 1, 521, 281, 521, 1, 31, 2207, 31, 1, 3571, 107, 3571, 1, 9349, 2161, 9349, 1, 211, 13201, 211, 1, 64079, 1103, 64079, 1, 15251, 90481, 15251, 1, 5779, 14503, 5779, 1, 1149851, 2521

Offset: 1

Alexander Adamchuk, Aug 16 2006, Sep 21 2006

a(1) = 1 and a(4k+2) = 1 for k>0.

For k >1 a(4k-1) = a(4k+1) = A072183(2k+1) = A061447(2k+1) Primitive part of Lucas(n).

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121709, A090585.

Cf. A061447, A072183.

Table[Numerator[Sum[Fibonacci[k],{k,1,n}]/Product[Fibonacci[k],{k,1,n}]],{n,1,100}]
With[{fibs=Fibonacci[Range[60]]},Numerator[Accumulate[fibs]/Rest[ FoldList[ Times,1,fibs]]]] (* This is significantly faster than the first program above *) (* Harvey P. Dale, Aug 19 2012 *)

a(n) = numerator( sum(k=1..n, Fibonacci(k)) / prod(k=1..n, Fibonacci(k)) ).

A072271 A partial product representation of f(n) = A015523(n) and L(n) = A072263(n).

Original entry on oeis.org

3, 1, 24, 19, 431, 14, 7589, 311, 5559, 241, 2345179, 286, 41223001, 4229, 70051, 95471, 12736968311, 5309, 223887209309, 88321, 21607111, 1306469, 69176042380099, 94846, 2821250547551, 22964761, 160204320879, 27289081, 375703599163598591, 119641

Offset: 1

Miklos Kristof, Jul 09 2002

nonn

For even n, f(n) = Product_{d|n} a(d); for odd n, f(n) = Product_{d|n} a(2d).
For odd prime p, a(p) = L(p)/3, where L(n) = 5*f(n-1) + f(n+1).
a(1)=3, a(2)=1.
a(2p) = f(p) for odd primes p.
a(2^(k+1)) = L(2^k).
a(3*2^k) = L(2^k) - 5^k.
For odd n, L(n) = Product_{d|n} a(d).
For k > 0 and odd n, L(n*2^k) = Product_{d|n} a(d*2^(k+1)).

			f(12) = a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 3*1*24*19*14*286 = 5477472 for even n;
f(7) = a(2)*a(14) = 1*4229 = 4229 for odd n.
L(6) = a(4)*a(12) = 19*286 = 5434 = 5*f(5) + f(7) = 5*241 + 4229 for even n;
L(15) = a(1)*a(3)*a(5)*a(15) = 3*24*431*70051 = 2173822632 for odd n.

Cf. A072270, A072280, A072183, A127259, A127607.

a(n) = (h-3)^g(n) * K(n, h^2/5) for n > 2 where h = (3+sqrt(29))/2, Phi(n, x) = n-th cyclotomic polynomial and g(n) is the order of Phi(n, x).

More terms and entry revised by Sean A. Irvine, Sep 19 2024

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A072270 A partial product representation of A006131 and A072265.

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

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A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

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A072271 A partial product representation of f(n) = A015523(n) and L(n) = A072263(n).

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