A078522 -id:A078522 - cp's OEIS frontend

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.

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

A111766 Numbers occurring in three Pythagorean triples of the form: odd: a, (a^2-1)/2, (a^2+1)/2 or even: a, a^2/4-1, a^2/4+1.

Original entry on oeis.org

0, 5, 24, 145, 840, 4901, 28560, 166465, 970224, 5654885, 32959080, 192099601, 1119638520, 6525731525, 38034750624, 221682772225, 1292061882720, 7530688524101, 43892069261880, 255821727047185, 1491038293021224, 8690408031080165, 50651409893459760

Offset: 1

Jeremy C. Buchanan (jbuchanan(AT)myhww.org), Nov 21 2005

This parallels Cassini's identity for Fibonacci numbers (Mathworld).

			a(5) = P(4)*P(6) = 12*70 = 840 = P(5)-1 = 29^2-1.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A076218, A078522 (bisections).

concat(0, Vec(-x^2*(-5+x)/((1+x)*(1-6*x+x^2)) + O(x^30))) \\ Colin Barker, Nov 04 2016

a(n) = A000129(n-1)*A000129(n+1) = A000129(n)^2 + (-1)^n.
G.f. -x^2*(-5+x) / ( (1+x)*(1-6*x+x^2) ). - R. J. Mathar, Sep 21 2011
a(n) =  A001333(n)^2 - A000129(n)^2 for n >= 1. - Richard R. Forberg, Aug 24 2013
From Colin Barker, Nov 04 2016: (Start)
a(n) = (6*(-1)^n+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/8 for n>0.
a(n) = 5*a(n-1)+5*a(n-2)-a(n-3) for n>3.
(End)

cp's OEIS Frontend

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.

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A111766 Numbers occurring in three Pythagorean triples of the form: odd: a, (a^2-1)/2, (a^2+1)/2 or even: a, a^2/4-1, a^2/4+1.

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cp's OEIS Frontend

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

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Crossrefs

Programs

Mathematica

Formula

Extensions

A111766 Numbers occurring in three Pythagorean triples of the form: odd: a, (a^2-1)/2, (a^2+1)/2 or even: a, a^2/4-1, a^2/4+1.

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PARI

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.