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A228873 F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045. 3
6, 30, 240, 1560, 10920, 74256, 510510, 3495030, 23965920, 164237040, 1125770256, 7715953440, 52886430870, 362487682830, 2484530961360, 17029219589256, 116720030923320, 800010932051760, 5483356663145790, 37583485265670630, 257601041359736256 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Mohanty and Mohanty prove in Corollary 2.5 that these numbers are Pythagorean. The number a(n) is primitive Pythagorean if F(n) and F(n+1) have opposite parity. Every third number, starting at a(1) = 6, is not primitive Pythagorean.

Since a(n) = F(n+1)*F(n+2)*(F(n+2)^2 - F(n+1)^2), a(n) is in A073120. - Robert Israel, Apr 06 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

FORMULA

G.f.: -6*x/((x-1)*(x^2+3*x+1)*(x^2-7*x+1)). - Alois P. Heinz, Oct 02 2013

a(n+5) = 5*a(n+4)+15*a(n+3)-15*a(n+2)-5*a(n+1)+a(n). - Robert Israel, Apr 06 2015

a(n) = 2 * A000217(A059840(n+2)). - Diego Rattaggi, Jan 27 2021

MAPLE

seq(mul(combinat:-fibonacci(i), i=n..n+3), n=1..30); # Robert Israel, Apr 06 2015

MATHEMATICA

Table[Fibonacci[n] Fibonacci[n+1] Fibonacci[n+2] Fibonacci[n+3], {n, 25}]

CoefficientList[Series[-6/((x - 1) (x^2 + 3 x + 1) (x^2 - 7 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2013 *)

Times@@@Partition[Fibonacci[Range[30]], 4, 1] (* Harvey P. Dale, Dec 23 2013 *)

LinearRecurrence[{5, 15, -15, -5, 1}, {6, 30, 240, 1560, 10920}, 30] (* Harvey P. Dale, Jul 24 2021 *)

PROG

(MAGMA) [Fibonacci(n)*Fibonacci(n+1)*Fibonacci(n+2)*Fibonacci(n+3): n in [1..30]]; // Vincenzo Librandi, Oct 04 2013

CROSSREFS

Cf. A000045 (Fibonacci numbers), A228874 (similar sequence for Lucas numbers).

Cf. A009112 (Pythagorean numbers), A024365, A073120.

Sequence in context: A259820 A126751 A009689 * A133668 A121772 A270845

Adjacent sequences:  A228870 A228871 A228872 * A228874 A228875 A228876

KEYWORD

nonn,changed

AUTHOR

T. D. Noe, Sep 24 2013

STATUS

approved

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Last modified August 5 01:31 EDT 2021. Contains 346456 sequences. (Running on oeis4.)