A103975 -id:A103975 - cp's OEIS frontend

A104008 Area of (m,m+1,m+1)-integer triangle for m in A103975.

120, 25080, 4890480, 949077360, 184120982760, 35718589344360, 6929223155685600, 1344233586759971040, 260774386791383159640, 50588886806491889178840, 9813983266108156439587920, 1903862164738670605583434320, 369339445976042880280120411080

Offset: 1

Zak Seidov, Feb 24 2005

Index entries for linear recurrences with constant coefficients, signature (208, -2718, 208, -1).

Cf. A103975.

LinearRecurrence[{208,-2718,208,-1},{120,25080,4890480,949077360},20] (* Harvey P. Dale, Mar 06 2017 *)

a(n) = (m/4)*sqrt((3*m+2)*(m+2)), where m=A103975(n).

G.f.: 120*x*(x+1) / ((x^2-194*x+1)*(x^2-14*x+1)). - Colin Barker, Apr 10 2013

More terms from Max Alekseyev, May 31 2007

More terms from Colin Barker, Apr 10 2013

A103974 Smaller sides (a) in (a,a,a+1)-integer triangle with integer area.

Original entry on oeis.org

1, 5, 65, 901, 12545, 174725, 2433601, 33895685, 472105985, 6575588101, 91586127425, 1275630195845, 17767236614401, 247465682405765, 3446752317066305, 48007066756522501, 668652182274248705

Offset: 1

Zak Seidov, Feb 23 2005

Corresponding areas are: 0, 12, 1848, 351780, 68149872, 13219419708, 2564481115560 (see A104009).
What is the next term? Is the sequence finite? The possible last two digits of "a" are (it may help in searching for more terms): {01, 05, 09, 15, 19, 25, 29, 33, 35, 39, 45, 49, 51, 55, 59, 65, 69, 75, 79, 83, 85, 89, 95, 99}.
Equivalently, positive integers a such that 3/16*a^4 + 1/4*a^3 - 1/8*a^2 - 1/4*a - 1/16 is a square (A000290), a direct result of Heron's formula. Conjecture: lim_{n->oo} a(n+1)/a(n) = 7 + 4*sqrt(3) (= 7 + A010502). - Rick L. Shepherd, Sep 04 2005
Values x^2 + y^2, where the pair (x, y) solves for x^2 - 3y^2=1, i.e., a(n)= (A001075(n))^2 + (A001353(n))^2 = A055793(n) + A098301(n). - Lekraj Beedassy, Jul 13 2006
Floretion Algebra Multiplication Program, FAMP Code: 1lestes[ 3'i - 2'j + 'k + 3i' - 2j' + k' - 4'ii' - 3'jj' + 4'kk' - 'ij' - 'ji' + 3'jk' + 3'kj' + 4e ]

Christian Aebi, and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.
Project Euler, Problem 94: Almost Equilateral Triangles.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A011922, A007655, A001353, A102341, A103975, A016064, A011945, A010502 (4*sqrt(3)), A000290 (square numbers), A350916.

A:=rsolve({-A(n+3)+15*A(n+2)-15*A(n+1)+A(n), A(0) = 1, A(1) = 5, A(2)=65}, A(n), makeproc); # Mihailovs

f[n_] := Simplify[((2 + Sqrt[3])^(2n) + (2 - Sqrt[3])^(2n) + 1)/3]; Table[ f[n], {n, 0, 16}] (* Or *)
a[1] = 1; a[2] = 5; a[3] = 65; a[n_] := a[n] = 15a[n - 1] - 15a[n - 2] + a[n - 3]; Table[ a[n], {n, 17}] (* Or *)
CoefficientList[ Series[(1 - 10x + 5x^2)/(1 - 15x + 15x^2 - x^3), {x, 0, 16}], x] (* Or *)
Range[0, 16]! CoefficientList[ Simplify[ Series[(E^x + E^((7 + 4Sqrt[3])x) + E^((7 - 4Sqrt[3])x))/3, {x, 0, 16}]], x] (* Robert G. Wilson v, Mar 24 2005 *)

for(a=1,10^6, b=a; c=a+1; s=(a+b+c)/2; if(issquare(s*(s-a)*(s-b)*(s-c)), print1(a,","))) /* Uses Heron's formula */ \\ Rick L. Shepherd, Sep 04 2005

Composite of comments from Alec Mihailovs (alec(AT)mihailovs.com) and David Terr, Mar 07 2005: (Start)
"a(n)^2 = A011922(n)^2 + (4*A007655(n))^2, so that A011922(n) = 1/2 base of triangles, A007655(n) = 1/4 height of triangles (conjectured by Paul Hanna).
Area is (a+1)/4*sqrt((3*a+1)*(a-1)). If a is even, the numerator is odd and the area is not an integer. That means a=2*k-1. In this case, Area=k*sqrt((3*k-1)*(k-1)).
Solving equation (3*k-1)*(k-1)=y^2, we get k=(2+sqrt(1+3*y^2))/3. That means that 1+3*y^2=x^2 with integer x and y. This is a Pell equation, all solutions of which have the form x=((2+sqrt(3))^n+(2-sqrt(3))^n)/2, y=((2+sqrt(3))^n-(2-sqrt(3))^n)/(2*sqrt(3)). Therefore k=(x+2)/3 is an integer only for even n. Then a=2*k-1=(2*x+1)/3 with even n. Q.E.D.
a(n)=(1/3)*((2+sqrt(3))^(2*n-2)+(2-sqrt(3))^(2*n-2)+1).
Recurrence: a(n+3)=15*a(n+2)-15*a(n+1)+a(n), a(0)=1, a(1)=5, a(2)=65.
G.f.: x*(1-10*x+5*x^2)/(1-15*x+15*x^2-x^3).
E.g.f.: 1/3*(exp(x)+exp((7+4*sqrt(3))*x)+exp((7-4*sqrt(3))*x)).
a(n) = 4U(n)^2 + 1, where U(1) = 0, U(2)=1 and U(n+1) = 4U(n) - U(n-1) for n>1. (U(n), V(n)) is the n-th solution to Pell's equation 3U(n)^2 + 1 = V(n)^2. (U(n) is the sequence A001353.)" (End)
a(n+1) = A098301(n+1) + A055793(n+2) - Creighton Dement, Apr 18 2005
a(n) = floor((7+4*sqrt(3))*a(n-1))-4, n>=3. - Rick L. Shepherd, Sep 04 2005
a(n)= [1+14*A007655(n+2)-194*A007655(n+1)]/3. - R. J. Mathar, Nov 16 2007
For n>=3, a(n) = 14*a(n-1) - a(n-2) - 4. It is one of 10 second-order linear recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916. - Max Alekseyev, Jan 22 2022

More terms from Creighton Dement, Apr 18 2005

Edited by Max Alekseyev, Jan 22 2022

A016064 Smallest side lengths of almost-equilateral Heronian triangles (sides are consecutive positive integers, area is a nonnegative integer).

Original entry on oeis.org

1, 3, 13, 51, 193, 723, 2701, 10083, 37633, 140451, 524173, 1956243, 7300801, 27246963, 101687053, 379501251, 1416317953, 5285770563, 19726764301, 73621286643, 274758382273, 1025412242451, 3826890587533, 14282150107683, 53301709843201, 198924689265123, 742397047217293

Offset: 0

Robert G. Wilson v

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.
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]
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
"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
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
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
a(0) = 1 corresponds to the degenerate triangle [1,2,3], with area = 0. - Wesley Ivan Hurt, May 20 2020
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
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
Then, in this case, with a(n) = m, the corresponding area of this triangle is 3 * A011945(n+1). - Bernard Schott, Mar 21 2023

			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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 100 terms from Zak Seidov)
David Eugene Smith and Yoshio Mikami, A history of Japanese mathematics, Dover, 2004, p. 168.
Eric Weisstein's World of Mathematics, Heronian Triangle.
Wikipedia, Heronian triangle.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A011945 (areas), A334277 (perimeters) A001353 (inradius).
Cf. A003500 (middle side lengths), this sequence (smallest side lengths), A335025 (largest side lengths).
Cf. A001353, A019973 (2 + sqrt(3)), A102341, A103974, A103975.
Cf. A072221.

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

LinearRecurrence[{5,-5,1},{1,3,13},26] (* Ray Chandler, Jan 27 2014 *)
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 *)
a[ n_] := 2 ChebyshevT[n, 2] - 1; (* Michael Somos, May 11 2019 *)

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

a(n)=if(n<1,1,-1+ceil((2+sqrt(3))^(n))) \\ Ralf Stephan

is(n)=issquare(3*n^2+6*n-9) \\ Charles R Greathouse IV, May 16 2014

{a(n) = 2 * polchebyshev(n, 1, 2) - 1}; /* Michael Somos, May 11 2019 */

a(n) = 3 + floor((2 + sqrt(3))*a(n-1)), n >= 3. - Rick L. Shepherd, Sep 04 2005
From Paul Barry, Feb 17 2004: (Start)
a(n) = 4*a(n-1) - a(n-2) + 2.
a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n - 1.
a(n) = 2*A001075(n) - 1.
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)
For n >= 1, a(n) = ceiling((2 + sqrt(3))^n) - 1.
a(n) = A003500(n) - 1. - T. D. Noe, Jun 17 2004
a(n) = [x^n] ( 1 + 2*x + sqrt(1 + 2*x + 3*x^2) )^n. - Peter Bala, Jun 23 2015
E.g.f.: exp((2 + sqrt(3))*x) + exp((2 - sqrt(3))*x) - exp(x). - Franck Maminirina Ramaharo, Nov 12 2018
a(n) = a(-n) for all integer n. - Michael Somos, May 11 2019

More terms from Rick L. Shepherd, Feb 18 2007

Definition revised by N. J. A. Sloane, Feb 04 2021

A011945 Areas of almost-equilateral Heronian triangles (integral side lengths m-1, m, m+1 and integral area).

Original entry on oeis.org

0, 6, 84, 1170, 16296, 226974, 3161340, 44031786, 613283664, 8541939510, 118973869476, 1657092233154, 23080317394680, 321467351292366, 4477462600698444, 62363009058485850, 868604664218103456, 12098102289994962534, 168504827395711372020, 2346969481249964245746

Offset: 1

E. K. Lloyd

Corresponding m's are in A016064. Corresponding values of lesser side give A016064.

Vincenzo Librandi, Table of n, a(n) for n = 1..890
Tanya Khovanova, Recursive Sequences
E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.
Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
P. Yiu, Heron triangles with consecutive sides, Recreational Mathematics, Chap. 9.3, pp. 80/360. (This is a download of 360 pages.)
Index entries for linear recurrences with constant coefficients, signature (14,-1).

Equals 6 * A007655(n+1).
Cf. this sequence (areas), A334277 (perimeters).
Cf. A003500 (middle side lengths), A016064 (smallest side lengths), A335025 (largest side lengths).
Cf. A102341, A103974, A103975.

CoefficientList[Series[6 x/(1 - 14 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{14,-1},{0,6},20] (* Harvey P. Dale, Jan 24 2015 *)

s(n) = floor((a+1)/4)*sqrt(3*(a+3)*(a-1)), where a = A016064(n). - Zak Seidov, Feb 23 2005
a(n) = 14*a(n-1) - a(n-2); a(1) = 0, a(2) = 6.
G.f.: 6*x^2/(1 - 14*x + x^2). - Philippe Deléham, Nov 17 2008
a(n) = (s/4)*((7 + 4*s)^n - (7 - 4*s)^n), where s = sqrt(3). - Zak Seidov, Apr 02 2014
E.g.f.: 6 - exp(7*x)*(12*cosh(4*sqrt(3)*x) - 7*sqrt(3)*sinh(4*sqrt(3)*x))/2. - Stefano Spezia, Dec 12 2022

Entry revised by N. J. A. Sloane, Feb 03 2007

A381336 a(n) is the smallest k > 0 for which a nondegenerate integer-sided triangle (k, k + n, c >= k + n) with an integer area exists.

Original entry on oeis.org

3, 6, 9, 12, 12, 18, 5, 7, 4, 24, 14, 36, 15, 10, 36, 14, 7, 8, 6, 21, 8, 3, 12, 5, 10, 15, 12, 20, 46, 35, 9, 28, 20, 14, 25, 16, 15, 12, 22, 21, 19, 16, 12, 6, 20, 5, 4, 10, 11, 20, 21, 30, 96, 24, 13, 9, 18, 7, 25, 63, 21, 18, 22, 9, 35, 9, 25, 21, 36, 17, 13

Offset: 1

Felix Huber, Mar 16 2025

nonn

Longest sides c are in A381337.

			a(5) = 12 because the nondegenerate integer-sided triangle (12, 12 + 5, 25 >= 12 + 5) has an integer area (90), and there is no smaller k > 0 than 12 that satisfies this condition.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heron's Formula.

Cf. A103974, A103975, A188158, A379830, A381337.

A381336:=proc(n)
    local k,c,s;
    for k do
        for c from k+n to 2*k+n-1 do
            s:=(n+2*k+c)/2;
            if issqr(s*(s-k)*(s-k-n)*(s-c)) then
                return k
            fi
        od
    od;
end proc;
seq(A381336(n),n=1..71);

cp's OEIS Frontend

A104008 Area of (m,m+1,m+1)-integer triangle for m in A103975.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A103974 Smaller sides (a) in (a,a,a+1)-integer triangle with integer area.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A016064 Smallest side lengths of almost-equilateral Heronian triangles (sides are consecutive positive integers, area is a nonnegative integer).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

PARI

Formula

Extensions

A011945 Areas of almost-equilateral Heronian triangles (integral side lengths m-1, m, m+1 and integral area).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A381336 a(n) is the smallest k > 0 for which a nondegenerate integer-sided triangle (k, k + n, c >= k + n) with an integer area exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple