A228209 -id:A228209 - cp's OEIS frontend

A224419 Numbers n such that triangular(n) + triangular(2*n) is a square.

0, 1, 25, 216, 1849, 36481, 311904, 2666689, 52606009, 449765784, 3845364121, 75857828929, 648561949056, 5545012396225, 109386936710041, 935225880773400, 7995904029992761, 157735886878050625, 1348595071513294176, 11530088066237165569, 227455039491212291641, 1944673157896289428824, 16626378995609962758169, 327990009210441246496129

Offset: 1

Alex Ratushnyak, Apr 18 2013

nonn

8 of the first 10 terms are of the form x^y. The two exceptions are a(7) = 311904 = 2^5 * 3^3 * 19^2 and a(10) = 449765784 = 2^3 * 3^5 * 13^2 * 37^2.
The corresponding squares are given by A075873(2*n-1)^2. E.g., triangular(a(10)) + triangular(2*a(10)) = 711142146^2 = A075873(19)^2.
Locations of squares in A147875, equivalent to solving the Diophantine equation n*(5*n+3)=2*s^2. - R. J. Mathar, Apr 19 2013

Max Alekseyev, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1442,-1442,0,-1,1).

Cf. A000217, A075873.

Cf. A220186 (numbers n such that triangular(2*n) - triangular(n) is a square).

LinearRecurrence[{1,0,1442,-1442,0,-1,1},{0,1,25,216,1849,36481,311904},30]  (* Harvey P. Dale, Jan 23 2015 *)

import math
for i in range(1<<30):
        s = i*(i+1)/2 + i*(2*i+1)
        t = int(math.sqrt(s))
        if s == t*t:  print(i)

a(n) = (A228209(2*n-1) - 3) / 10. - Max Alekseyev, Sep 04 2013

G.f.: x^2*(x+1)*(x^4 + 23*x^3 + 168*x^2 + 23*x + 1) / (x^6 - 1442*x^3 + 1) / (1-x). - Max Alekseyev, Sep 04 2013

Terms a(11) onward from Max Alekseyev, Sep 04 2013

A343034 Positive numbers m such that m^2 with last digit z deleted is still a perfect square k^2, and z divides m-k.

Original entry on oeis.org

1, 13, 19, 487, 721, 18493, 27379, 702247, 1039681, 26666893, 39480499, 1012639687, 1499219281, 38453641213, 56930852179, 1460225726407, 2161873163521, 55450123962253, 82094249361619, 2105644484839207, 3117419602578001, 79959040299927613, 118379850648602419, 3036337886912410087

Offset: 1

Bernard Schott, Apr 03 2021

This sequence is the answer to the problem A1872 proposed on French mathematical site Diophante (see link).
Equivalent to the two Diophantine equations: m^2 = 10*k^2 + z and m-k = q*z for some q >= 1.
There exist solutions iff z = 1 or z = 9.
When (m, k, z) is a solution, then (19m+60k, 6m+19k, z) is another solution.
There is only one solution such that z = m-k: (13, 4, 9), see 1st example.
There exist two distinct families of solutions corresponding to z = 1 and z = 9, odd indices correspond to z = 1 and even indices to z = 9.
-> For z = 1, all solutions m of Pell equation m^2 - 10*k^2 = 1 are terms because z = 1 divides every m-k.
   First few solutions (m, k) are (1, 0), (19, 6), (721, 228), (27379, 86568), ... with m = A078986(q) and corresponding k = 6*A078987(q).
-> For z = 9, solutions m must satisfy m^2 - 10*k^2 = 9 with 9 divides m-k. Among the 3 fundamental solutions (3, 0), (7, 2), (13, 4) of Pell equation m^2 -10*k^2 = 9, only (13, 4) gives solutions where 9 divides m-k.
   First few solutions (m, k) are (13, 4), (487, 154), (18493, 5848), ... with m = A228209(3q).

			For m = 13, 13^2 = 169, 4^2 = 16, 13^2 - 10*4^2 = 9 and 9 = 13-4 divides 13-4.
For m = 19, 19^2 = 361, 6^2 = 36, 19^2 - 10*6^2 = 1 and 1 divides 19-6 = 13.
For m = 487, 487^2 = 237169, 154^2 = 23716, 487^2 - 10*154^2 = 9 and 9 divides 487-154 = 333 = 9*37.

Diophante, A1872, Carrément magiques (in French).
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Subsequence of A031149.
A078986 is a subsequence.
Cf. A078987, A228209.

LinearRecurrence[{0, 38, 0, -1}, {1, 13, 19, 487}, 24] (* Amiram Eldar, Apr 03 2021 *)

a(2n+1) = A078986(n) for n >= 0.
a(2n) = A228209(3n) for n >= 1.
a(n+4) = 38*a(n+2) - a(n), a(1) = 1, a(2) = 13, a(3)= 19, a(4) = 487.
G.f.: x*(1 + 13*x - 19*x^2 - 7*x^3)/(1 - 38*x^2 + x^4). - Stefano Spezia, Apr 03 2021

cp's OEIS Frontend

A224419 Numbers n such that triangular(n) + triangular(2*n) is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions