cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A159951 Fibonacci integral quotients associated with the dividends in A159950 and the divisors in A003481.

Original entry on oeis.org

12, 856800, 139890541190400, 50664770469826998541056000, 40527253814267058837705250384270510080000, 71554565901386985191123530075861409411081105273676595200000
Offset: 1

Views

Author

Enoch Haga, Apr 27 2009

Keywords

Comments

The first example of an integral quotient in the Fibonacci sequence is 12 because 240/20=12. 240 is the product of terms through 8, and 20 the sum. Thereafter, with every other additional pair of terms in the Fibonacci sequence, another integral quotient occurs.
Let m be an even positive integer. Then the sequence defined by b_m(n) = Product_{k = 1..2*n+1} F(m*k) / Sum_{k = 1..2*n+1} F(m*k) appears to be integral. - Peter Bala, Nov 12 2021

Examples

			The first two integral quotients occur in the Fibonacci sequence as illustrated by the following: (1*1*2*3*5*8)/(1+1+2+3+5+8) = 240/20 = 12, integral; (1*1*2*3*5*8*13*21*34*55)/(1+1+2+3+5+8+13+21+34+55) = 122522400/143 = 856800, integral.

Crossrefs

Cf. A000045, A001519, A001906, A003481, A033890, A159950, A175553.

Programs

  • Maple
    with(combinat):
seq(mul(fibonacci(k), k = 1..4*n+2)/(fibonacci(4*n+4) - 1), n = 1..10); # Peter Bala, Nov 04 2021
  • UBASIC
    10 'Fibo 20 'R=SUM:S=PRODUCT 30 'T integral every other pair 40 A=1:S=1:print A;:S=S*1 50 B=1:print B;:S=S*B 60 C=A+B:print C;:R=R+C:S=S*C 70 D=B+C:print D;:R=R+D:R=R+2:print R:S=S*D:print S 80 T=S/R:if T=int(S/R) then print T:stop 90 A=C:B=D:R=R-2:goto 60

Formula

a(n) = (Product_{k = 1..4*n+2} Fibonacci(k))/(Sum_{k = 1..4*n+2} Fibonacci(k)) = (Product_{k = 1..4*n+2} Fibonacci(k))/(Fibonacci(4*n+4) - 1) = Fibonacci(2*n+1)/Fibonacci(2*n+3) * Product_{k = 1..4*n+1} Fibonacci(k), which shows a(n) is integral. Cf. A175553. - Peter Bala, Nov 11 2021

A081006 a(n) = Fibonacci(4n) - 1, or Fibonacci(2n+1)*Lucas(2n-1).

Original entry on oeis.org

2, 20, 143, 986, 6764, 46367, 317810, 2178308, 14930351, 102334154, 701408732, 4807526975, 32951280098, 225851433716, 1548008755919, 10610209857722, 72723460248140, 498454011879263, 3416454622906706, 23416728348467684, 160500643816367087
Offset: 1

Views

Author

R. K. Guy, Mar 01 2003

Keywords

Comments

Apart from the offset, the same as A003481. - R. J. Mathar, Sep 18 2008

References

  • Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Links

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Programs

  • GAP
    List([1..30], n-> Fibonacci(4*n)-1); # G. C. Greubel, Jul 15 2019
  • Magma
    [Fibonacci(4*n)-1: n in [1..30]]; // Vincenzo Librandi, Apr 15 2011
  • Maple
    with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n)-1) od # James Sellers, Mar 03 2003
  • Mathematica
    Fibonacci[4*Range[30]]-1 (* or *) LinearRecurrence[{8,-8,1}, {2,20,143}, 30] (* Harvey P. Dale, Mar 19 2018 *)
  • PARI
    vector(30, n, fibonacci(4*n)-1) \\ G. C. Greubel, Jul 15 2019
  • Sage
    [fibonacci(4*n)-1 for n in (1..30)] # G. C. Greubel, Jul 15 2019

Formula

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: x*(2+4*x-x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012

Extensions

More terms from James Sellers, Mar 03 2003

A159950 Dividends where Fibonacci products/sums yield integral quotients.

Original entry on oeis.org

240, 122522400, 137932073613734400, 342696507457909818131702784000, 1879127177606120717127879344567470740879360000, 22740756589119797763590969093409514524935686067027158720512000000
Offset: 1

Views

Author

Enoch Haga, Apr 27 2009

Keywords

Comments

In looking at the Fibonacci sequence I happened to notice that after each pair of terms >1 the product of terms divided by the sum of terms produced an integral quotient every other time. Example 240/20=12, integral.

Examples

			This table illustrates the alternating nature of the first three integral quotients: 1 1 2 3 -- 6/7=.85+ 5 8 -- 240/20=12 Integral 13 21 -- 65520/54=1213.33+ 34 55 -- 122522400/143=856800 Integral 89 144 -- 1570247078400/376=4176189038.29+ 233 377 -- 137932073613734400/986=139890541190400 Integral etc.

Links

Crossrefs

Cf. A159951, A001519, A001906, A003481, A033890, A003266, A052449.

Programs

  • Maple
    seq(mul(fibonacci(k), k = 1..4*n+2), n = 1..10); # Peter Bala, Nov 04 2021
  • UBASIC
    10 'Fibo 20 'R=SUM:S=PRODUCT 30 'T integral every other pair 40 A=1:S=1:print A;:S=S*1 50 B=1:print B;:S=S*B 60 C=A+B:print C;:R=R+C:S=S*C 70 D=B+C:print D;:R=R+D:R=R+2:print R:S=S*D:print S 80 T=S/R:if T=int(S/R) then print T:stop 90 A=C:B=D:R=R-2:goto 60

Formula

a(1)=240 because in the Fibonacci sequence up to 8 : 1 1 2 3 5 8, the product is 240 1*1*2*3*5*8. The sum is 1+1+2+3+5+8=20 (see A003481). The integral quotient is 12. From then on, every other pair produces an integral quotient.
a(n) = Product_{k = 1..4*n+2} Fibonacci(k) = A003266(4*n+2) = A052449(4*n+2) - 1. - Peter Bala, Nov 04 2021
Showing 1-3 of 3 results.

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