A075873 -id:A075873 - cp's OEIS frontend

A075836 Numbers k such that 10*k^2 + 9 is a square.

0, 2, 4, 18, 80, 154, 684, 3038, 5848, 25974, 115364, 222070, 986328, 4380794, 8432812, 37454490, 166354808, 320224786, 1422284292, 6317101910, 12160109056, 54009348606, 239883517772, 461763919342, 2050932962736

Offset: 1

Gregory V. Richardson, Oct 14 2002

(5/4)*a(n)^2 +1 is a triangular number. - Bruno Berselli, Aug 17 2013

A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (0,0,38,0,0,-1).

Cf. A221874.

I:=[0,2,4,18,80,154]; [n le 6 select I[n] else 38*Self(n-3)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 17 2013

CoefficientList[Series[2 x (1 + 2 x + 9 x^2 + 2 x^3 + x^4) / (1 - 38 x^3 + x^6), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 17 2013 *)

From Gregory V. Richardson, Oct 16 2002: (Start)
Limit_{n->oo} a(n)/a(n-3) = 19 + 6*sqrt(10).
Limit_{n->oo} a(3*n)/a(3*n-1) = (11 + 2*sqrt(10))/9.
Limit_{n->oo} a(3*n+1)/a(3*n) = (7 + 2*sqrt(10))/3.
Limit_{n->oo} a(3*n+2)/a(3*n+1) = (7 + 2*sqrt(10))/3. (End)
G.f.: 2*x^2*(1+2*x+9*x^2+2*x^3+x^4) / ( 1-38*x^3+x^6 ). - R. J. Mathar, Jul 03 2011
a(n) = 2*A075873(n). - R. J. Mathar, Jul 03 2011

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

Original entry on oeis.org

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

cp's OEIS Frontend

A075836 Numbers k such that 10*k^2 + 9 is a square.

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