A228873 a(n) = F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045.

6, 30, 240, 1560, 10920, 74256, 510510, 3495030, 23965920, 164237040, 1125770256, 7715953440, 52886430870, 362487682830, 2484530961360, 17029219589256, 116720030923320, 800010932051760, 5483356663145790, 37583485265670630, 257601041359736256

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, The Fibonacci Quarterly, Vol. 28, No. 1 (1990), pp. 31-42.

H.-J. Seiffert, Problem B-1020, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 44, No. 4 (2006), p. 278; Two Fibonacci Identities, Solution to Problem B-1020 by Harris Kwong, ibid., Vol. 45, No. 2 (2007), pp. 182-184.

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

Sum_{n>=1} 1/a(n) = (12-5*sqrt(5))/4. - Diego Rattaggi, Aug 16 2021

a(n) = 3 * Sum_{k=1..n} 2^(n-k)*F(k)^2*F(k+1)*F(k+2) (Seiffert, 2006). - Amiram Eldar, Jan 11 2022

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 * A366809 A133668 A121772

Adjacent sequences: A228870 A228871 A228872 * A228874 A228875 A228876

Keyword

nonn

Author

T. D. Noe, Sep 24 2013

Status

approved