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A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3.

%I A206351 #57 Jun 28 2025 22:11:23
%S A206351 1,3,16,105,715,4896,33553,229971,1576240,10803705,74049691,507544128,
%T A206351 3478759201,23843770275,163427632720,1120149658761,7677619978603,
%U A206351 52623190191456,360684711361585,2472169789339635
%N A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3.
%C A206351 A Pell sequence related to Heronian triangles (rational triangles), see A206334. The connection is this: consider the problem of finding triangles with area a positive integer n, and with sides (a, b, n) where a, b are rational. Note that n is both the area and one side. For many values of n this is not possible, and the sequence of such numbers n is quite erratic (see A206334). Nonetheless, each term in this sequence is such a value of n. For example, for n = 105 you can take the other two sides, a and b, to be 10817/104, and 233/104 and the area will equal n, i.e., 105.
%H A206351 Reinhard Zumkeller, <a href="/A206351/b206351.txt">Table of n, a(n) for n = 1..1000</a>
%H A206351 Pridon Davlianidze, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2020.pdf">Problem B-1279</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 4 (2020), p. 368; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2021.pdf">An Unusual Generalization</a>, Solution to Problem B-1279 by J. N. Senadheera, ibid., Vol. 59, No. 4 (2021), pp. 370-371.
%H A206351 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A206351 From _Bruno Berselli_, Feb 07 2012: (Start)
%F A206351 G.f.: x*(1-5*x)/(1-8*x+8*x^2-x^3).
%F A206351 a(n) = A081018(n-1) + 1. (End)
%F A206351 a(n) = -A003482(-n) = Fibonacci(2*n)*Fibonacci(2*n-3). - _Michael Somos_, Jun 26 2018
%F A206351 a(n) = A089508(n-1) + 2 for n>1. - _Bruno Berselli_, Jun 20 2019 [Formula found by Umberto Cerruti]
%F A206351 Product_{n>=2} (1 - 1/a(n)) = 1/phi (A094214) (Davlianidze, 2020). - _Amiram Eldar_, Nov 30 2021
%F A206351 a(n) = (Fibonacci(2*n-2) + 1/Lucas(2*n-2))*(Fibonacci(2*n-1) + 1/Lucas(2*n-1)). - _Peter Bala_, Sep 03 2022
%e A206351 G.f. = x + 3*x^2 + 16*x^3 + 105*x^4 + 715*x^5 + 4896*x^6 + 33553*x^7 + ... - _Michael Somos_, Jun 26 2018
%p A206351 genZ := proc(n)
%p A206351 local start;
%p A206351 option remember;
%p A206351     start := [1, 3];
%p A206351     if n < 3 then start[n]
%p A206351     else 7*genZ(n - 1) - genZ(n - 2) - 4
%p A206351     end if
%p A206351 end proc:
%p A206351 seq(genZ(n),n=1..20);
%t A206351 LinearRecurrence[{8, -8, 1}, {1, 3, 16}, 50] (* _Charles R Greathouse IV_, Feb 07 2012 *)
%t A206351 RecurrenceTable[{a[1] == 1, a[2] == 3, a[n] == 7 a[n - 1] - a[n - 2] - 4}, a, {n, 20}] (* _Bruno Berselli_, Feb 07 2012 *)
%t A206351 a[ n_] := Fibonacci[2 n] Fibonacci[2 n - 3]; (* _Michael Somos_, Jun 26 2018 *)
%t A206351 nxt[{a_,b_}]:={b,7b-a-4}; NestList[nxt,{1,3},20][[;;,1]] (* _Harvey P. Dale_, Aug 29 2024 *)
%o A206351 (PARI) Vec((1-5*x)/(1-8*x+8*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Feb 07 2012
%o A206351 (PARI) {a(n) = fibonacci(2*n) * fibonacci(2*n - 3)}; /* _Michael Somos_, Jun 26 2018 */
%o A206351 (Haskell)
%o A206351 a206351 n = a206351_list !! (n-1)
%o A206351 a206351_list = 1 : 3 : map (subtract 4)
%o A206351                (zipWith (-) (map (* 7) (tail a206351_list)) a206351_list)
%o A206351 -- _Reinhard Zumkeller_, Feb 08 2012
%o A206351 (Magma) [Fibonacci(2*n)*Fibonacci(2*n-3): n in [1..30]]; // _G. C. Greubel_, Aug 12 2018
%Y A206351 Cf. A000045, A003482, A089508, A094214.
%Y A206351 Subsequence of A206334.
%K A206351 nonn,easy
%O A206351 1,2
%A A206351 _James R. Buddenhagen_, Feb 06 2012