A206334 -id:A206334 - cp's OEIS frontend

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

1, 3, 16, 105, 715, 4896, 33553, 229971, 1576240, 10803705, 74049691, 507544128, 3478759201, 23843770275, 163427632720, 1120149658761, 7677619978603, 52623190191456, 360684711361585, 2472169789339635

Offset: 1

James R. Buddenhagen, Feb 06 2012

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.

			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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Pridon Davlianidze, Problem B-1279, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 4 (2020), p. 368; An Unusual Generalization, Solution to Problem B-1279 by J. N. Senadheera, ibid., Vol. 59, No. 4 (2021), pp. 370-371.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A003482, A089508, A094214.

Subsequence of A206334.

a206351 n = a206351_list !! (n-1)
a206351_list = 1 : 3 : map (subtract 4)
               (zipWith (-) (map (* 7) (tail a206351_list)) a206351_list)
-- Reinhard Zumkeller, Feb 08 2012

[Fibonacci(2*n)*Fibonacci(2*n-3): n in [1..30]]; // G. C. Greubel, Aug 12 2018

genZ := proc(n)
local start;
option remember;
    start := [1, 3];
    if n < 3 then start[n]
    else 7*genZ(n - 1) - genZ(n - 2) - 4
    end if
end proc:
seq(genZ(n),n=1..20);

LinearRecurrence[{8, -8, 1}, {1, 3, 16}, 50] (* Charles R Greathouse IV, Feb 07 2012 *)
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 *)
a[ n_] := Fibonacci[2 n] Fibonacci[2 n - 3]; (* Michael Somos, Jun 26 2018 *)
nxt[{a_,b_}]:={b,7b-a-4}; NestList[nxt,{1,3},20][[;;,1]] (* Harvey P. Dale, Aug 29 2024 *)

Vec((1-5*x)/(1-8*x+8*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 07 2012

{a(n) = fibonacci(2*n) * fibonacci(2*n - 3)}; /* Michael Somos, Jun 26 2018 */

From Bruno Berselli, Feb 07 2012: (Start)
G.f.: x*(1-5*x)/(1-8*x+8*x^2-x^3).
a(n) = A081018(n-1) + 1. (End)
a(n) = -A003482(-n) = Fibonacci(2*n)*Fibonacci(2*n-3). - Michael Somos, Jun 26 2018
a(n) = A089508(n-1) + 2 for n>1. - Bruno Berselli, Jun 20 2019 [Formula found by Umberto Cerruti]
Product_{n>=2} (1 - 1/a(n)) = 1/phi (A094214) (Davlianidze, 2020). - Amiram Eldar, Nov 30 2021
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

A228914 Positive integers N such that 1/N = p/q - q/p + r/s - s/r for some positive integers p,q,r,s.

Original entry on oeis.org

4, 5, 9, 12, 15, 20, 21, 22, 24, 26, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 43, 44, 53, 55, 56, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 74, 76, 77, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 102, 103, 104, 105, 106, 108, 109, 110, 112, 113, 115, 117, 122, 123, 129, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141, 143, 144, 147

Offset: 1

Thomas Bokk, Sep 13 2013

nonn

Positive integer N belongs to this sequence if and only if the elliptic curve y^2 = x^3 + (8*N^2+1)*x^2 + 16*N^4*x has positive rank.

Cf. A206334, A228380

{ isA228914(n) = ellanalyticrank(ellinit([0, 8*n^2+1, 0, 16*n^4, 0]))[1]; } /* Max Alekseyev, Dec 30 2015 */

More terms from Max Alekseyev, Sep 13 2013

A257642 Positive integers N such that there is a triangle with rational sides having area and perimeter both equal N.

Original entry on oeis.org

21, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 39, 42, 43, 45, 47, 50, 51, 52, 55, 56, 58, 60, 61, 62, 63, 64, 66, 67, 71, 74, 75, 76, 77, 79, 81, 83, 85, 86, 88, 90, 91, 93, 94, 95, 96, 98, 99, 100, 102, 103, 105, 106, 107, 108, 109, 110, 112, 113, 115, 116, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 137, 138, 141, 143, 145, 147, 148, 149, 150

Offset: 1

Thomas Bokk, Nov 05 2015

nonn

A positive integer N is in the sequence if and only if there exist positive rational numbers x,y such that x*y>1 and 4*x*y*(x+y)/(x*y-1)=N.

Except for N=27, a positive integer N is in this sequence if and only if N>20 and the elliptic curve w^2 = u^3 + N^2*(u+64)^2 has positive rank.

Cf. A135412, A206334, A228914, A117319, A228380.

cp's OEIS Frontend

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

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A228914 Positive integers N such that 1/N = p/q - q/p + r/s - s/r for some positive integers p,q,r,s.

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A257642 Positive integers N such that there is a triangle with rational sides having area and perimeter both equal N.

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