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.
%I A228873 #37 Jan 05 2025 19:51:40
%S A228873 6,30,240,1560,10920,74256,510510,3495030,23965920,164237040,
%T A228873 1125770256,7715953440,52886430870,362487682830,2484530961360,
%U A228873 17029219589256,116720030923320,800010932051760,5483356663145790,37583485265670630,257601041359736256
%N A228873 a(n) = F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045.
%C A228873 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.
%C A228873 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
%H A228873 Vincenzo Librandi, <a href="/A228873/b228873.txt">Table of n, a(n) for n = 1..1000</a>
%H A228873 Supriya Mohanty and S. P. Mohanty, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/28-1/mohanty.pdf">Pythagorean Numbers</a>, The Fibonacci Quarterly, Vol. 28, No. 1 (1990), pp. 31-42.
%H A228873 H.-J. Seiffert, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/August2006elementary.pdf">Problem B-1020</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 44, No. 4 (2006), p. 278; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/elementary45-2.pdf">Two Fibonacci Identities</a>, Solution to Problem B-1020 by Harris Kwong, ibid., Vol. 45, No. 2 (2007), pp. 182-184.
%H A228873 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).
%F A228873 G.f.: -6*x/((x-1)*(x^2+3*x+1)*(x^2-7*x+1)). - _Alois P. Heinz_, Oct 02 2013
%F A228873 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
%F A228873 a(n) = 2 * A000217(A059840(n+2)). - _Diego Rattaggi_, Jan 27 2021
%F A228873 Sum_{n>=1} 1/a(n) = (12-5*sqrt(5))/4. - _Diego Rattaggi_, Aug 16 2021
%F A228873 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
%p A228873 seq(mul(combinat:-fibonacci(i),i=n..n+3),n=1..30); # _Robert Israel_, Apr 06 2015
%t A228873 Table[Fibonacci[n] Fibonacci[n+1] Fibonacci[n+2] Fibonacci[n+3], {n, 25}]
%t A228873 CoefficientList[Series[-6/((x - 1) (x^2 + 3 x + 1) (x^2 - 7 x + 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 04 2013 *)
%t A228873 Times@@@Partition[Fibonacci[Range[30]],4,1] (* _Harvey P. Dale_, Dec 23 2013 *)
%t A228873 LinearRecurrence[{5,15,-15,-5,1},{6,30,240,1560,10920},30] (* _Harvey P. Dale_, Jul 24 2021 *)
%o A228873 (Magma) [Fibonacci(n)*Fibonacci(n+1)*Fibonacci(n+2)*Fibonacci(n+3): n in [1..30]]; // _Vincenzo Librandi_, Oct 04 2013
%Y A228873 Cf. A000045 (Fibonacci numbers), A228874 (similar sequence for Lucas numbers).
%Y A228873 Cf. A009112 (Pythagorean numbers), A024365, A073120.
%K A228873 nonn
%O A228873 1,1
%A A228873 _T. D. Noe_, Sep 24 2013