A164055 -id:A164055 - cp's OEIS frontend

A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.

0, 3, 18, 105, 612, 3567, 20790, 121173, 706248, 4116315, 23991642, 139833537, 815009580, 4750223943, 27686334078, 161367780525, 940520349072, 5481754313907, 31950005534370, 186218278892313, 1085359667819508, 6325939728024735, 36870278700328902, 214895732473948677

Offset: 1

Pierre CAMI, Apr 29 2005

The ratio k(n) /(2*j(n)) tends to sqrt(2) as n increases.
The squares of the numbers in this sequence are one less than a triangular number: a(n)^2 = A164080(n). For example, 18^2 is 324, and 325 is a triangular number. a(n)^2 + 1 = A164055(n). a(n)^2 = A072221(n)(A072221(n)+1)/2 - 1. - Tanya Khovanova & Alexey Radul, Aug 09 2009
For n > 0, a(n+1) is the n-th almost balancing number of first type (see Tekcan and Erdem). - Stefano Spezia, Nov 25 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
Ahmet Tekcan and Alper Erdem, General Terms of All Almost Balancing Numbers of First and Second Type, arXiv:2211.08907 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A001109, A005319, A072221, A103328, A164080, A164055.

a106328 n = a106328_list !! (n-1)
a106328_list = 0 : 3 : zipWith (-) (map (* 6) (tail a106328_list)) a106328_list
-- Reinhard Zumkeller, Jan 10 2012

s=0;lst={};Do[s+=n;If[Sqrt[s-1]==Floor[Sqrt[s-1]],AppendTo[lst,Sqrt[s-1]]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Rest@ CoefficientList[Series[3 x^2/(1 - 6 x + x^2), {x, 0, 24}], x] (* Michael De Vlieger, Nov 02 2020 *)

concat(0, Vec(3*x^2/(1-6*x+x^2) + O(x^40))) \\ Michel Marcus, Sep 07 2016

a(n)=([0,1;-1,6]^n*[-3;0])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

a(1)=0, a(2)=3 then a(n) = 6*a(n-1) - a(n-2).
a(n) = ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1))*3/4/sqrt(2). - Max Alekseyev, Jan 11 2007
a(n) = 3*A001109(n). - M. F. Hasler, R. J. Mathar, Jun 03 2009
a(n) = (3/4)*A005319(n-1).
G.f.: 3*x^2/(1 - 6*x + x^2). - Philippe Deléham, Nov 17 2008
E.g.f.: 3 - 3*exp(3*x)*(4*cosh(2*sqrt(2)*x) - 3*sqrt(2)*sinh(2*sqrt(2)*x))/4. - Stefano Spezia, Nov 25 2022

More terms from Max Alekseyev, Jan 11 2007

A214838 Triangular numbers of the form k^2 + 2.

Original entry on oeis.org

3, 6, 66, 171, 2211, 5778, 75078, 196251, 2550411, 6666726, 86638866, 226472403, 2943171003, 7693394946, 99981175206, 261348955731, 3396416785971, 8878171099878, 115378189547778, 301596468440091, 3919462027838451, 10245401755863186, 133146330756959526, 348042063230908203

Offset: 1

Alex Ratushnyak, Mar 07 2013

Corresponding k values are in A077241.

Except 3, all terms are in A089982: in fact, a(2) = 3+3 and a(n) = (k-2)*(k-1)/2+(k+1)*(k+2)/2, where k = sqrt(a(n)-2) > 2 for n > 2. [Bruno Berselli, Mar 08 2013]

			2211 is in the sequence because 2211 = 47^2 + 2.

Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. A000217, A001110, A001219, A006454, A029549, A069017, A077241, A089982, A164055, A165892.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-3*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)))); // Bruno Berselli, Mar 08 2013

LinearRecurrence[{1, 34, -34, -1, 1}, {3, 6, 66, 171, 2211}, 25] (* Bruno Berselli, Mar 08 2013 *)

t[n]:=((5-2*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor(n/2))+(5+2*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor(n/2))-2)/4$
makelist(expand(t[n]*(t[n]+1)/2), n, 1, 25); /* Bruno Berselli, Mar 08 2013 */

for(n=1, 10^9, t=n*(n+1)/2; if(issquare(t-2), print1(t,", "))); \\ Joerg Arndt, Mar 08 2013

import math
for i in range(2, 1<<32):
      t = i*(i+1)//2 - 2
      sr = int(math.sqrt(t))
      if sr*sr == t:
          print(f'{sr:10} {i:10} {t+2}')

G.f.: -3*x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)). - Joerg Arndt, Mar 08 2013

a(n) = A000217(t), where t = ((5-2*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor(n/2))+(5+2*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor(n/2))-2)/4. - Bruno Berselli, Mar 08 2013

A226072 Primes p such that p-1 is a triangular number and p-2 is a square.

Original entry on oeis.org

2, 11, 11027, 16944049179227, 94511138700672573788068264540372768937231403134027

Offset: 1

Alex Ratushnyak, May 25 2013

nonn

Intersection of A129545 and A164055.
a(6) has 106 decimal digits: 12450353555...7512227.
a(7) has 381 decimal digits: 49394105798...1431227.
Roots of corresponding squares and triangular numbers: A226074 and A226073.

Cf. A226069-A226074, A129545, A164055.

import java.io.*;
import java.math.BigInteger;
public class A226072 {
public static void main (String[] args) throws Exception {
  try {
    BufferedReader in = new BufferedReader(
      new FileReader(new File("b164055.txt")));
    String line;
    while ((line = in.readLine()) != null) {
      BigInteger b = new BigInteger(line.split(" ")[1]);
      b = b.add(BigInteger.ONE);
      if (b.isProbablePrime(80))
        System.out.printf("%s, ", b.toString());
    }
  } catch (Exception e) {  e.printStackTrace();  }
}
}

Select[Prime[Range[1500]],OddQ[Sqrt[8(#-1)+1]]&&IntegerQ[Sqrt[#-2]]&] (* The program generates the first 3 terms of the sequence. *) (* Harvey P. Dale, Jul 20 2024 *)

A164080 Perfect squares one less than a triangular number.

Original entry on oeis.org

0, 9, 324, 11025, 374544, 12723489, 432224100, 14682895929, 498786237504, 16944049179225, 575598885856164, 19553418069930369, 664240615491776400, 22564627508650467249, 766533094678624110084, 26039560591564569275625

Offset: 1

Tanya Khovanova & Alexey Radul, Aug 09 2009

nonn

			324=18^2 is a perfect square and 325=A000217(25) is a triangular number. Therefore 324 is in this sequence.

Index entries for linear recurrences with constant coefficients, signature (35, -35, 1).

LinearRecurrence[{35,-35,1},{0,9,324},20] (* Harvey P. Dale, Oct 24 2023 *)

a(n) = A164055(n)-1.
a(n) = A072221(n)*(A072221(n)+1)/2 - 1.
a(n) = 35*a(n-1) -35*a(n-2) +a(n-3) = 9*A001110(n-1). G.f.: 9*x^2*(1+x)/((1-x)*(x^2-34*x+1)). [R. J. Mathar, Oct 21 2009]

Comments turned into formulas. - R. J. Mathar, Oct 21 2009

A328791 Triangular numbers of the form k^2 + 3.

Original entry on oeis.org

3, 28, 903, 30628, 1040403, 35343028, 1200622503, 40785822028, 1385517326403, 47066803275628, 1598885794044903, 54315050194251028, 1845112820810490003, 62679520857362409028, 2129258596329511416903, 72332112754346025765628, 2457162575051435364614403

Offset: 1

Jon E. Schoenfield, Oct 27 2019

nonn

There exist triangular numbers of the form k^2 + j for j=0 (A001110), j=1 (A164055), j=2 (A214838), and j=3 (this sequence), but not for j=4,7,8,13,16,18,... (A328792).

Cf. A001110, A164055, A214838, A328792.
Intersection of A000217 and A117950.
Cf. A276598 (the k's).

limit = 10**7 # rough limit for k
A000217 = set(k*(k+1)//2 for k in range(14*limit//10))
A117950 = set(k**2 + 3 for k in range(limit))
print(sorted(A000217 & A117950)) # Michael S. Branicky, Mar 28 2021

a(1) = 3, a(2) = 28; for n > 2, a(n) = 34*a(n-1) - a(n-2) - 46.

A328792 Numbers that are not the difference between any triangular number and the largest square that does not exceed it.

Original entry on oeis.org

4, 7, 8, 13, 16, 18, 22, 23, 25, 26, 31, 33, 34, 37, 38, 40, 43, 47, 48, 49, 52, 58, 59, 60, 61, 63, 64, 67, 68, 70, 73, 76, 79, 81, 83, 85, 86, 88, 92, 93, 94, 97, 98, 99, 102, 103, 106, 108, 112, 113, 114, 115, 118, 121, 123, 124, 125, 130, 133, 134, 138

Offset: 1

Jon E. Schoenfield, Oct 27 2019

nonn

			For any triangular number t, let f(t) = t - floor(sqrt(t))^2.
0 is not a term: for each term t in A001110, f(t) = 0.
1 is not a term: for each term t > 1 in A164055, f(t) = 1.
2 is not a term: for each term t in A214838, f(t) = 2.
3 is not a term: for each term t > 3 in A328791, f(t) = 3.
4 is a term, however: there exists no triangular number t such that f(t) = 4.

The complement of A230044.

Cf. A001110, A064784, A164055, A214838, A328791.

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A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.

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A214838 Triangular numbers of the form k^2 + 2.

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A226072 Primes p such that p-1 is a triangular number and p-2 is a square.

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A164080 Perfect squares one less than a triangular number.

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A328791 Triangular numbers of the form k^2 + 3.

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A328792 Numbers that are not the difference between any triangular number and the largest square that does not exceed it.

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