A016064 - OEIS

%I #84 Nov 05 2023 19:07:14

%S 1,3,13,51,193,723,2701,10083,37633,140451,524173,1956243,7300801,

%T 27246963,101687053,379501251,1416317953,5285770563,19726764301,

%U 73621286643,274758382273,1025412242451,3826890587533,14282150107683,53301709843201,198924689265123,742397047217293

%N Smallest side lengths of almost-equilateral Heronian triangles (sides are consecutive positive integers, area is a nonnegative integer).

%C Least side in a triangle with integer sides (m, m+1, m+2) (m >= 1) and integer area. The degenerate triangle with sides (1,2,3) is included.

%C Also describes triangles whose sides are consecutive integers and in which the inscribed circle has an integer radius. - _Harvey P. Dale_, Dec 28 2000 [Then, the length of this inradius is A001353(n). - _Bernard Schott_, Mar 21 2023]

%C Equivalently, positive integers m such that (3/16)*m^4 + (3/4)*m^3 + (3/8)*m^2 - (3/4)*m - 9/16 is a square (A000290), a direct result of Heron's formula. - _Rick L. Shepherd_, Sep 04 2005

%C "The problem is to find the sides of a triangle that shall have the values n, n + 1, and n + 2 and such that the perpendicular upon the longest side from the opposite vertex shall be rational. Nakane solves it as follows..." (Smith and Mikami, 2004). - _Jonathan Sondow_, May 09 2013

%C For n >= 1 all terms are congruent to {1,3} mod 10. Among first 100 terms there are 6 prime numbers: 3, 13, 193, 37633, 7300801, 1416317953. - _Zak Seidov_, Jun 14 2018

%C n > 1 is in this sequence if and only if the triangle with sides 4, n, n+2 has integer area (compare with A072221 for sides 3, n, n+1). - _Michael Somos_, May 11 2019

%C a(0) = 1 corresponds to the degenerate triangle [1,2,3], with area = 0. - _Wesley Ivan Hurt_, May 20 2020

%C Since this is a list it should really have offset 1, but that would require a large number of changes. - _N. J. A. Sloane_, Feb 04 2021

%C Least distance from centroid of a triangle to vertices, distances being m, m+1, m+2 and triangle area being a nonnegative integer. - _Alexandru Petrescu_, Feb 28 2023

%C Then, in this case, with a(n) = m, the corresponding area of this triangle is 3 * A011945(n+1). - _Bernard Schott_, Mar 21 2023

%D Nakane Genkei (Nakane the Elder), Shichijo Beki Yenshiki, 1691.

%H Vincenzo Librandi, <a href="/A016064/b016064.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Zak Seidov)

%H David Eugene Smith and Yoshio Mikami, <a href="https://archive.org/details/ahistoryjapanes01mikagoog/page/n180">A history of Japanese mathematics</a>, Dover, 2004, p. 168.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).

%F a(n) = 3 + floor((2 + sqrt(3))*a(n-1)), n >= 3. - _Rick L. Shepherd_, Sep 04 2005

%F From _Paul Barry_, Feb 17 2004: (Start)

%F a(n) = 4*a(n-1) - a(n-2) + 2.

%F a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n - 1.

%F a(n) = 2*A001075(n) - 1.

%F G.f.: (1 - 2*x + 3*x^3)/((1 - x)*(1 - 4*x + x^2)) = (1 - 2*x + 3*x^2)/(1 - 5*x + 5*x^2 - x^3). (End)

%F For n >= 1, a(n) = ceiling((2 + sqrt(3))^n) - 1.

%F a(n) = A003500(n) - 1. - _T. D. Noe_, Jun 17 2004

%F a(n) = [x^n] ( 1 + 2*x + sqrt(1 + 2*x + 3*x^2) )^n. - _Peter Bala_, Jun 23 2015

%F E.g.f.: exp((2 + sqrt(3))*x) + exp((2 - sqrt(3))*x) - exp(x). - _Franck Maminirina Ramaharo_, Nov 12 2018

%F a(n) = a(-n) for all integer n. - _Michael Somos_, May 11 2019

%e G.f. = 1 + 3*x + 13*x^2 + 51*x^3 + 193*x^4 + 723*x^5 + 2701*x^6 + ... - _Michael Somos_, May 11 2019

%t LinearRecurrence[{5,-5,1},{1,3,13},26] (* _Ray Chandler_, Jan 27 2014 *)

%t CoefficientList[Series[(1 - 2 x + 3 x^2) / (1 - 5 x + 5 x^2 - x^3), {x, 0, 33}], x] (* _Vincenzo Librandi_, Nov 13 2018 *)

%t a[ n_] := 2 ChebyshevT[n, 2] - 1; (* _Michael Somos_, May 11 2019 *)

%o (PARI) for(a=1,10^9, b=a+1; c=a+2; s=(a+b+c)/2; if(issquare(s*(s-a)*(s-b)*(s-c)), print1(a,","))) \\ _Rick L. Shepherd_, Feb 18 2007

%o (PARI) a(n)=if(n<1,1,-1+ceil((2+sqrt(3))^(n))) \\ _Ralf Stephan_

%o (PARI) is(n)=issquare(3*n^2+6*n-9) \\ _Charles R Greathouse IV_, May 16 2014

%o (PARI) {a(n) = 2 * polchebyshev(n, 1, 2) - 1}; /* _Michael Somos_, May 11 2019 */

%o (Magma) I:=[1,3,13]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)+2: n in [1..30]]; // _Vincenzo Librandi_, Nov 13 2018

%Y Cf. A011945 (areas), A334277 (perimeters) A001353 (inradius).

%Y Cf. A003500 (middle side lengths), this sequence (smallest side lengths), A335025 (largest side lengths).

%Y Cf. A001353, A019973 (2 + sqrt(3)), A102341, A103974, A103975.

%Y Cf. A072221.

%K nonn,easy

%O 0,2

%A _Robert G. Wilson v_

%E More terms from _Rick L. Shepherd_, Feb 18 2007

%E Defition revised by _N. J. A. Sloane_, Feb 04 2021