A284876 -id:A284876 - cp's OEIS frontend

A046176 Indices of square numbers that are also hexagonal.

1, 35, 1189, 40391, 1372105, 46611179, 1583407981, 53789260175, 1827251437969, 62072759630771, 2108646576008245, 71631910824649559, 2433376321462076761, 82663163018885960315, 2808114166320660573949

Offset: 1

Eric W. Weisstein

Bisection (even part) of Chebyshev sequence with Diophantine property.
(3*b(n))^2 - 2*(2*a(n+1))^2 = 1 with companion sequence b(n) = A077420(n), n >= 0.
Sequence also refers to inradius of primitive Pythagorean triangles with consecutive legs, odd followed by even. - Lekraj Beedassy, Apr 23 2003
As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> oo} a(n)/a(n-1) = (1 + sqrt(2))^4 = 17 + 12*sqrt(2). - Ant King, Nov 08 2011
Integers of the form sqrt((m+1)*(2*m+1)). The corresponding values of m form A078522. Subsequence of A284876. - Jonathan Sondow, Apr 07 2017

M. Rignaux, Query 2175, L'Intermédiaire des Mathématiciens, 24 (1917), 80.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences.
E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=3.
Serge Perrine, About the diophantine equation z^2 = 32y^2 - 16, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT] (2008) , v_n in Theorem 5.6.
P. H. van der Kamp, Global classification of two-component..., Found. Comput. Math. 9 (5) (2009) 559-597 near Eq. (4.7).
Eric Weisstein's World of Mathematics, Hexagonal Square Number.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (34,-1).

Cf. A000129, A008844, A046177, A078522, A284876.

Cf. A001109, A001110 (partial sums).

List([0..20], n-> Lucas(2,-1, 4*n-2)[1]/2 ); # G. C. Greubel, Jan 13 2020

I:=[1,35]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

seq( simplify(ChebyshevU(2*(n-1), 3)), n = 1..20); # G. C. Greubel, Jan 13 2020

LinearRecurrence[{34, -1}, {1, 35}, 15] (* Ant King, Nov 08 2011 *)
Fibonacci[4*Range[20] -2, 2]/2 (* G. C. Greubel, Jan 13 2020 *)

vector(21, n, polchebyshev(2*(n-1), 2, 3) ) \\ G. C. Greubel, Jan 13 2020

[lucas_number1(4*n-2, 2,-1)/2 for n in (1..20)] # G. C. Greubel, Jan 13 2020

a(n) = 34*a(n-1) - a(n-2); a(0)=-1, a(1)=1.
a(n+1) = S(2*n, 6) = S(n, 34) + S(n-1, 34), n >= 1, with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(n, 34) = A029547(n).
G.f.: x*(1+x)/(1-34*x+x^2).
a(n+1) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*6^(2*(n-k)), n >= 0.
a(n) = A001109(2n+1). - Lekraj Beedassy, Apr 23 2003
Define f(x,s) = s*x + sqrt((s^2-1)*x^2+1); f(0,s)=0. a(n) = f(f(a(n-1),3),3). - Marcos Carreira, Dec 27 2006
From Antonio Alberto Olivares, Mar 22 2008: (Start)
a(n) = (sqrt(2)/8)*(3 + 2*sqrt(2))*(17 + 12*sqrt(2))^(n-1) - (sqrt(2)/8)*(3 - 2*sqrt(2))*(17 - 12*sqrt(2))^(n-1).
a(n) = (sqrt(2)/8)*( (17+12*sqrt(2))^(n-1/2) - (17-12*sqrt(2))^(n-1/2) ).
a(n) = (sqrt(2)/8)*( (3+2*sqrt(2))^(2n-1) - (3-2*sqrt(2))^(2n-1) ).
a(n) = (sqrt(2)/8)*( (1+sqrt(2))^(4n-2) - (1-sqrt(2))^(4n-2) ).
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). (End)
a(n+1) = 17*a(n) + 6*sqrt(8*a(n)^2+1) for n >= 0. - Richard Choulet, May 01 2009
a(n) = b such that (-1)^(n+1) * Integral_{x=-Pi/2..Pi/2} cos((2*n-1)*x)/(3-sin(x)) dx = c + b*log(2). - Francesco Daddi, Aug 01 2011
a(n) are the nonzero integer square roots of A227970. - Richard R. Forberg, Aug 01 2013
a(n) = y/5, where y are solutions to: y^2 = 2x^2 - x - 3. - Richard R. Forberg, Nov 24 2013
a(n) = sqrt((A078522(n)+1)*(2*A078522(n)+1)). - Jonathan Sondow, Apr 07 2017
a(n) = Pell(4*n-2)/2. - G. C. Greubel, Jan 13 2020
a(n) = A001653(n)*A002315(n). - Gerry Martens, Mar 23 2024

Chebyshev comments from Wolfdieter Lang, Nov 29 2002

A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

Original entry on oeis.org

1, 1, 1, 1, 25, 49, 18, 25, 32, 1, 841, 1681, 49, 169, 289, 50, 169, 288, 49, 289, 529, 128, 289, 450, 98, 625, 1152, 289, 625, 961, 800, 841, 882, 162, 1681, 3200, 288, 1369, 2450, 529, 1369, 2209, 1, 28561, 57121, 49, 5329, 10609, 961, 1681, 2401, 289, 2809, 5329

Offset: 1

Jonathan Sondow, Mar 31 2017

This is a 3-column table read by rows a, a+d, a+2*d. Each row has product a square. The rows are ordered by the products. The square roots of the products form A284876, which contains A046176. The pairs a,d form A284874.
Goldbach proved that a product of 3 consecutive positive integers is never a square.
Euler proved that a product of 4 consecutive positive integers is never a square.
Erdos and Selfridge (1975) proved that a product of 2 or more consecutive positive integers is never a square or a higher power.
Saradha (1998) proved that 18, 25, 32 is the only arithmetic progression a, a+d, ..., a+(k-1)*d whose product is a square if a>=1, 1=3 with gcd(a,d)=1. In 1997 she showed that the product is not a square or a higher power if a>=1, 1=3 with gcd(a,d)=1.
(1, 1+d, 1+2*d) is in the table if and only if d is in A078522. - Robert Israel, Apr 05 2017 - Jonathan Sondow, Apr 06 2017

			18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2 and gcd(18,25,32) = 1, so 18,25,32 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..1248 (triples with product < 10^18)
P. Erdős and J.L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math., 19 (1975), 292-301.
N. Saradha, On perfect powers in products with terms from arithmetic progressions, Acta Arith., 82 (1997), 147-172.
N. Saradha, Squares in products with terms in an arithmetic progression, Acta Arith., 86 (1998), 27-43.

Cf. A046176, A078522, A284874, A284876.

N:= 10^11: # to get all triples where the product <= N
Res:= [1,0]:
for a from 1 to floor(N^(1/3)) do
  for d from 1 while a*(a+d)*(a+2*d) <= N do
     if igcd(a,d) = 1 and issqr(a*(a+d)*(a+2*d)) then
       Res:= Res, [a,d]
     fi
  od
od:
Res:= sort([Res], (s,t) -> s[1]*(s[1]+s[2])*(s[1]+2*s[2]) <= t[1]*(t[1]+t[2])*(t[1]+2*t[2])):
map(t -> (t[1],t[1]+t[2],t[1]+2*t[2]), Res); # Robert Israel, Apr 05 2017

nn = 50000; t = {};
p[a_, b_, c_] := a *b*c; Do[
If[p[a, a + d, a + 2 d] <= 2 nn^2 && GCD[a, d] == 1 &&
   IntegerQ[Sqrt[p[a, a + d, a + 2 d]]],
  AppendTo[t, {a, a + d, a + 2 d}]], {a, 1, nn}, {d, 0, nn}];
Sort[t, p[#1[[1]], #1[[2]], #1[[3]]] <
    p[#2[[1]], #2[[2]], #2[[3]]] &] // Flatten

a(3*k+1) = A284874(2*k+1) and a(3*k+2) = A284874(2*k+1)+A284874(2*k+2) and a(3*k+3) = A284874(2*k+1)+2*A284874(2*k+2) and a(3*k+1)*a(3*k+2)*a(3*k+3) = A284876(k+1)^2 for k>=0.

A284874 List of pairs (a,d) of coprime integers a>0, d>=0 such that a(a+d)(a+2*d) is a square, ordered by the squares.

Original entry on oeis.org

1, 0, 1, 24, 18, 7, 1, 840, 49, 120, 50, 119, 49, 240, 128, 161, 98, 527, 289, 336, 800, 41, 162, 1519, 288, 1081, 529, 840, 1, 28560, 49, 5280, 961, 720, 289, 2520, 242, 3479, 49, 9360, 512, 3713, 529, 3696, 1568, 1241, 338, 6887, 2401, 1320, 2178, 2047

Offset: 1

Jonathan Sondow, Apr 04 2017

This is a 2-column table read by rows. For each row a,d the product a*(a+d)*(a+2*d) is a square. The rows are ordered by those products.
The main entry for this sequence is A284666, formed by the triples a, a+d, a+2*d. The square roots of the products a*(a+d)*(a+2*d) form A284876.
For a=1 the d values 0, 24, 840, 28560, ... are A078522.

			gcd(18,7)=1 and 18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2, so 18,7 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..832

Cf. A078522, A284666, A284876.

nn = 50000; t = {};
p[a_, d_] := a (a + d) (a + 2 d); Do[
If[p[a, d] <= 2 nn^2 && GCD[a, d] == 1 && IntegerQ[Sqrt[p[a, d]]],
  AppendTo[t, {a, d}]], {a, 1, nn}, {d, 0, nn}];
Sort[t, p[#1[[1]], #1[[2]]] < p[#2[[1]], #2[[2]]] &] // Flatten

a(2*k+1) = A284666(3*k+1) and a(2*k+2) = A284666(3*k+2)-A284666(3*k+1) and a(2*k+1)*[a(2*k+1)+a(2*k+2)]*[a(2*k+1)+2*a(2*k+2)] = A284876(k+1)^2 for k>=0.

a(37)-a(52) from Giovanni Resta, Apr 06 2017

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A046176 Indices of square numbers that are also hexagonal.

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A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

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Keywords

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Programs

Maple

Mathematica

Formula

A284874 List of pairs (a,d) of coprime integers a>0, d>=0 such that a*(a+d)*(a+2*d) is a square, ordered by the squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A284874 List of pairs (a,d) of coprime integers a>0, d>=0 such that a(a+d)(a+2*d) is a square, ordered by the squares.