A079896 -id:A079896 - cp's OEIS frontend

A261249 Number of classes of proper solutions of the Pell equation x^2 - D(n) y^2 = +4 for D(n) = A079896(n), n >= 1.

2, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Wolfdieter Lang, Sep 16 2015

nonn

See the W. Lang link on A225953, Table 2. References will also be found there. For the present class number see especially Theorem 109 pp. 207-208 of the Nagell reference.
These class numbers should not be confused with the class numbers of indefinite binary quadratic forms of discriminant D(n), which are given in A087048(n).
If a(n) = 2 then the proper positive fundamental solution for the second class [x2(n), y2(n)] is obtained from the solution of the first class [x1(n), y1(n)] (shown in the mentioned Table 2 under Pell(X, Y)) by application of the matrix M(n) = [[x0(n), D(n)*y0(n)], [y0(n), x0(n)]] on (x1(n), -y1(n))^T (T for transposed), where x0(n) and y0(n) is the positive (proper) fundamental solution of x^2 - D(n)*y^2 = +1 found under A033313 and A033317 for the appropriate D from A000037. Application of positive powers of M(n) to the proper positive fundamental solution of each class produces all positive solutions.
If a(n) = 1 the class is called ambiguous (see Nagell, p. 205). In this case the proper positive fundamental solution [x1(n), y1(n)] = [x(n), y(n)] and the negative one [x1(n), -y1(n)] belong to the same class.
For every D(n) = A079896(n) there is the improper positive fundamental solution [2*x0(n), 2*y0(n)].
Conjecture: For even D(n), i.e., D from 4*A000037, and a(n) = 0 one finds for r(n) = D(n)/4 coincidence with Conway's so-called rectangular numbers A007969. The first D values are 8, 20, 24, 40, 48, 52, 56, 68, 72, 80, ... This is equivalent to the conjecture that X^2 - r*y^2 = +1 has an even fundamental positive solution y = y0 precisely for the numbers A007969 (because x has to be even, x = 2*X, and whenever y0 is even all y solutions are even). See A261250 and A262024 for the y0 and x0 values, respectively.

			n=1: D(1) = 5 = A000037(3) with the a(1) = 2 proper positive fundamental solutions [x, y] = [3, 1] and [7, 3] for the two classes.
  [x0(1), y0(1)] = [A033313(3), A033317(3)] = [9, 4], and (7, 3)^T = [[9, 4*5], [4, 9]] (3, -1)^T.
  All other positive solutions in each of the two classes are obtained by applying positive powers of this matrix M(5) to the fundamental solutions.
  The improper positive fundamental solution is [2*9, 2*4] = [18, 8].
n=2: D(2) = 8 = A000037(6) has a(2) = 0, hence there are only the improper solutions obtainable from [2*3, 2*1] = [6, 2], the smallest positive one. For this even D one has, with x = 2*X, X^2 - 8/4 y^2 = +1, which has an even positive fundamental solution y0 = 2, and r(2) = D(2)/4 = 2 is A007969(1).

Nagell, T. Introduction to number theory, Chelsea Publishing Company, 1964, page 52.

Cf. A079896, A225953, A087048, A033313, A033317, A261250, A262024.

Offset corrected by Robin Visser, Jun 08 2025

A306638 a(n) is the norm of the fundamental unit of binary quadratic forms with discriminant D = A079896(n).

Original entry on oeis.org

-1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1

Offset: 1

Jianing Song, Mar 02 2019

sign

The fundamental unit of binary quadratic forms with discriminant D is the number (x_1 + (y_1)*sqrt(D))/2, where (x_1,y_1) is the smallest solution to x^2 - D*y^2 = +-4. Each term is either -1 or 1 depending on whether (x_1)^2 - D*(y_1)^2 = -4 or 4.
All solutions to x^2 - D*y^2 = +-4 are given by the identity (x_n + (y_n)*sqrt(D))/2 = ((x_1 + (y_1)*sqrt(D))/2)^n.
The discriminants D corresponding to (x_1)^2 - D*(y_1)^2 = -4 are listed in A226696.

			Fundamental units and their norms for the first 15 discriminants in the form (X + Y*sqrt(D))/2 (N = (X^2 - D*Y^2)/4 are the corresponding norms) are:
   D |  X |  Y |  N
   5 |  1 |  1 | -1
   8 |  2 |  1 | -1
  12 |  4 |  1 |  1
  13 |  3 |  1 | -1
  17 |  8 |  2 | -1
  20 |  4 |  1 | -1
  21 |  5 |  1 |  1
  24 | 10 |  2 |  1
  28 | 16 |  3 |  1
  29 |  5 |  1 | -1
  32 |  6 |  1 |  1
  33 | 46 |  8 |  1
  37 | 12 |  2 | -1
  40 |  6 |  1 | -1
  41 | 64 | 10 | -1

D. A. Buell, Binary Quadratic Forms, Springer, 1989, Sections 3.2 and 3.3, pp. 31-48.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A079896, A226696.

A014077 is a subsequence listing the corresponding values for only fundamental discriminants (A003658).

using Nemo
function b(D)
    for j in 1:10000
        issquare(D*j^2 - 4) && return -1
        issquare(D*j^2 + 4) && return 1
    end
0 end
F = findall(n -> ZZ(n) % 4 <= 1 && !issquare(ZZ(n)), 1:100)
map(n -> b(ZZ(n)), F) |> println # Peter Luschny, Mar 08 2019

b(D) = for(n=1, oo, if(issquare(D*n^2-4), return(-1)); if(issquare(D*n^2+4), return(1)))
for(n=2, 200, if(n%4 <= 1 && !issquare(n), print1(b(n), ", ")))

a(n) = -1 if D = A079896(n) is in A226696, otherwise 1.

Offset changed to 1 by Robin Visser, Jun 09 2025

A257161 The length of the period under Zagier-reduction of the principal indefinite quadratic binary form of discriminant D(n) = A079896(n).

Original entry on oeis.org

1, 2, 1, 3, 5, 4, 1, 2, 2, 5, 1, 4, 7, 6, 11, 3, 1, 2, 10, 7, 2, 7, 1, 11, 9, 8, 2, 4, 21, 7, 1, 2, 4, 9, 6, 21, 2, 3, 1, 27, 11, 10, 3, 5, 17, 6, 23, 16, 1, 2, 8, 11, 2, 15, 2, 6, 2, 27, 1

Offset: 1

Barry R. Smith, Apr 16 2015

nonn

A binary quadratic form A*x^2 + B*x*y + C*y^2 with integer coefficients A, B, and C and positive discriminant D = B^2 - 4*A*C is Zagier-reduced if A>0, C>0, and B>A+C. (This differs from the classical reduced forms defined by Lagrange.)  There are finitely many Zagier-reduced forms of given discriminant.
Zagier defines a reduction operation on binary quadratic forms with positive discriminants, which permutes the reduced forms.  The reduced forms are thereby partitioned into disjoint cycles.
There is a unique Zagier-reduced form with A=1 for each discriminant in A079896. The cycle containing this form is the principal cycle.  a(n) is the length of this cycle for the discriminant D=A079896(n).

			For n=4, the a(4) = 3 forms in the principal cycle of discriminant A079896(4) = 13 are x^2 + 5*x*y + 3*y^2, 3*x^2 + 5*x*y + y^2, and 3*x^2 + 7*x*y + 3*y^2.

D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.

Cf. A226166.

With D=n^2-4, a(n) equals the number of pairs (a,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), a > (sqrt(D) - k)/2, a exactly dividing (D-k^2)/4.

Offset corrected by Robin Visser, Jun 08 2025

A000037 Numbers that are not squares (or, the nonsquares).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

N. J. A. Sloane, Simon Plouffe

Note the remarkable formula for the n-th term (see the FORMULA section)!
These are the natural numbers with an even number of divisors. The number of divisors is odd for the complementary sequence, the squares (sequence A000290) and the numbers for which the number of divisors is divisible by 3 is sequence A059269. - Ola Veshta (olaveshta(AT)my-deja.com), Apr 04 2001
a(n) is the largest integer m not equal to n such that n = (floor(n^2/m) + m)/2. - Alexander R. Povolotsky, Feb 10 2008
Union of A007969 and A007970; A007968(a(n)) > 0. - Reinhard Zumkeller, Jun 18 2011
Terms of even numbered rows in the triangle A199332. - Reinhard Zumkeller, Nov 23 2011
If a(n) and a(n+1) are of the same parity then (a(n)+a(n+1))/2 is a square. - Zak Seidov, Aug 13 2012
Theaetetus of Athens proved the irrationality of the square roots of these numbers in the 4th century BC. - Charles R Greathouse IV, Apr 18 2013
4*a(n) are the even members of A079896, the discriminants of indefinite binary quadratic forms. - Wolfdieter Lang, Jun 14 2013

			For example note that the squares 0, 1, 4, 9, 16 are not included.

Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 58-60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 9900 terms from N. J. A. Sloane)
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
A. J. dos Reis and D. M. Silberger, Generating nonpowers by formula, Math. Mag., 63 (1990), 53-55.
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011.
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
S. Kaji, T. Maeno, K. Nuida, and Y. Numata, Polynomial Expressions of Carries in p-ary Arithmetics, arXiv preprint arXiv:1506.02742 [math.CO], 2015-2016.
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458. doi 10.2307/2308078, see example 4 (includes the formula). [Nicolas Normand (Nicolas.Normand(AT)polytech.univ-nantes.fr), Nov 24 2009]
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343-347.
R. D. Nelson, Sequences which omit powers, The Mathematical Gazette, Number 461, 1988, pages 208-211.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Rosetta Code, Sequence of non-squares
J. Scholes, 27th Putnam 1966 Prob. A4
Aaron Snook, Augmented Integer Linear Recurrences, 2012. - From _N. J. A. Sloane_, Dec 19 2012
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Continued Fraction

Cf. A007412, A000005, A000290, A059269, A134986, A087153, A172151, A000196, A049068 (subsequence).
Cf. A242401 (subsequence).
Cf. A086849 (partial sums), A048395.

a000037 n = n + a000196 (n + a000196 n)
-- Reinhard Zumkeller, Nov 23 2011

[n : n in [1..1000] | not IsSquare(n) ];

at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;

Maple
```
A000037 := n->n+floor(1/2+sqrt(n));
```

a[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[a[n], {n, 71}] (* Robert G. Wilson v, Sep 24 2004 *)
With[{upto=100},Complement[Range[upto],Range[Floor[Sqrt[upto]]]^2]] (* Harvey P. Dale, Dec 02 2011 *)
a[ n_] :=  If[ n < 0, 0, n + Round @ Sqrt @ n]; (* Michael Somos, May 28 2014 *)

A000037(n):=n + floor(1/2 + sqrt(n))$ makelist(A000037(n),n,1,50); /* Martin Ettl, Nov 15 2012 */

{a(n) = if( n<0, 0, n + (1 + sqrtint(4*n)) \ 2)};

from math import isqrt
def A000037(n): return n+isqrt(n+isqrt(n)) # Chai Wah Wu, Mar 31 2022

from math import isqrt
def A000037(n): return n+(k:=isqrt(n))+int(n>=k*(k+1)+1) # Chai Wah Wu, Jun 17 2024

a(n) = n + floor(1/2 + sqrt(n)).
a(n) = n + floor(sqrt( n + floor(sqrt n))).
A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 26 2010
A173517(a(n)) = n; a(n)^2 = A030140(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = A000194(n) + n. - Jaroslav Krizek, Jun 14 2009
a(A002061(n)) = a(n^2-n+1) = A002522(n) = n^2 + 1. - Jaroslav Krizek, Jun 21 2009

Edited by Charles R Greathouse IV, Oct 30 2009

A014601 Numbers congruent to 0 or 3 mod 4.

Original entry on oeis.org

0, 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124

Offset: 0

Eric Rains (rains(AT)caltech.edu)

Discriminants of orders in imaginary quadratic fields (negated). [Comment corrected by Christopher E. Thompson, Dec 11 2016]
Numbers such that Langford-Skolem problem has a solution - see A014552.
Complement of A042963. - Reinhard Zumkeller, Oct 04 2004
Also called skew amenable numbers; a number k is skew amenable if there exist a set {a(i)} of integers satisfying the relations k = Sum_{i=1..k} a(i) = -Product_{i=1..k} a(i). Thus we have 8 = 1 + 1 + 1 + 1 + 1 + 1 - 2 + 4 = -(1*1*1*1*1*1*(-2)*4). - Lekraj Beedassy, Jan 07 2005
Possible nonpositive discriminants of quadratic equation a*x^2 + b*x + c or discriminants of binary quadratic forms a*x^2 + b*x*y + c*y^2. - Artur Jasinski, Apr 28 2008
Also, disregarding the 0 term, positive integers m such that, equivalently,
  (i) +-1 +-2 +-... +-m is even for all choices of signs,
  (ii) +-1 +-2 +-... +-m = 0 for some choices of signs,
  (iii) for all -m <= k <= m, k = +-1 +-2 +-... +-(k-1) +-(k+1) +-(k+2) +-... +-m for at least one choice of signs. - Rick L. Shepherd, Oct 29 2008
A145768(a(n)) is even. - Reinhard Zumkeller, Jun 05 2012
Multiples of 4 interleaved with 1 less than multiples of 4. - Wesley Ivan Hurt, Nov 08 2013
((2*k+0) + (2*k+1) + ... + (2*k+m-1) + (2*k+m)) is even if and only if m = a(n) for some n where k is any nonnegative integer. - Gionata Neri, Jul 24 2015
Numbers whose binary reflected Gray code (A014550) ends with 0. - Amiram Eldar, May 17 2021

			G.f. = 3*x + 4*x^2 + 7*x^3 + 8*x^4 + 11*x^5 + 12*x^6 + 15*x^7 + 16*x^8 + ...

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 108.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. F. Barger, Solution to problem 10454: Amenable Numbers, Amer. Math. Monthly, Vol. 105, No. 4 (April 1998), p. 368.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Heiko Harborth, Solution of Steinhaus's problem with plus and minus signs, Journal of Combinatorial Theory, Series A, Volume 12, Issue 2 (March 1972), Pages 253-259.
Mickaël Launay, Les routes de Numland, L’énigme maths du "Monde" n°2. In French.
Michael Penn, The most beautiful arrangement of numbers, YouTube video, 2024.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004676, A014550, A042948, A079896.

Cf. A274406. - Bruno Berselli, Jun 26 2016

a014601 n = a014601_list !! n
a014601_list = [x | x <- [0..], mod x 4 `elem` [0, 3]]
-- Reinhard Zumkeller, Jun 05 2012

[n: n in [0..200]|n mod 4 in {0,3}]; // Vincenzo Librandi, Dec 24 2010

A014601:=n->3*n-2*floor(n/2); seq(A014601(k), k=0..100); # Wesley Ivan Hurt, Nov 08 2013

aa = {}; Do[Do[Do[d = b^2 - 4 a c; If[d <= 0, AppendTo[aa, -d]], {a, 0, 50}], {b, 0, 50}], {c, 0, 50}]; Union[aa] (* Artur Jasinski, Apr 28 2008 *)
Select[Range[0, 124], Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] (* Ant King, Nov 18 2010 *)
CoefficientList[Series[2 x/(1 - x)^2 + (1/(1 - x) + 1/(1 + x)) x/2, {x, 0, 100}], x] (* Vincenzo Librandi, May 18 2014 *)
a[ n_] := 2 n + Mod[n, 2]; (* Michael Somos, Jul 24 2015 *)

{a(n) = 2*n + n%2}; /* Michael Somos, Dec 27 2010 */

a(n) = (n + 1)*2 + 1 - n mod 2. - Reinhard Zumkeller, Apr 21 2003
A014494(n) = A000217(a(n)). - Reinhard Zumkeller, Oct 04 2004
a(n) = Sum_{k=1..n} (2 - (-1)^k). - William A. Tedeschi, Mar 20 2008
A139131(a(n)) = A078636(a(n)). - Reinhard Zumkeller, Apr 10 2008
From R. J. Mathar, Sep 25 2009: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2.
G.f.: x*(3+x)/((1+x)*(x-1)^2). (End)
a(n) = 2*n + (n mod 2). - Paolo Valzasina (p.valzasina(AT)gmail.com), Nov 24 2009
a(n) = (4*n - (-1)^n + 1)/2. - Bruno Berselli, Oct 06 2010
a(n) = 4*n - a(n-1) - 1 (with a(0) = 0). - Vincenzo Librandi, Dec 24 2010
a(n) = -A042948(-n) for all n in Z. - Michael Somos, Dec 27 2010
G.f.: 2*x / (1 - x)^2 + (1 / (1 - x) + 1 / (1 + x)) * x/2. - Michael Somos, Dec 27 2010
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0) = 3 and b(k) = 2^(k+1) for k > 0. - Philippe Deléham, Oct 17 2011
a(n) = ceiling((4/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
a(n) = 3n - 2*floor(n/2). - Wesley Ivan Hurt, Nov 08 2013
a(n) = A042948(n+1) - 1 for all n in Z. - Michael Somos, Jul 24 2015
a(n) + a(n+1) = A004767(n) for all n in Z. - Michael Somos, Jul 24 2015
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)/4 - Pi/8. - Amiram Eldar, Dec 05 2021
E.g.f.: ((4*x + 1)*exp(x) - exp(-x))/2. - David Lovler, Aug 04 2022

A028884 a(n) = (n + 3)^2 - 8.

Original entry on oeis.org

1, 8, 17, 28, 41, 56, 73, 92, 113, 136, 161, 188, 217, 248, 281, 316, 353, 392, 433, 476, 521, 568, 617, 668, 721, 776, 833, 892, 953, 1016, 1081, 1148, 1217, 1288, 1361, 1436, 1513, 1592, 1673, 1756, 1841, 1928, 2017, 2108, 2201, 2296, 2393

Offset: 0

Patrick De Geest

From Klaus Purath, Jan 04 2023: (Start)
The product of two consecutive terms belongs to the sequence: a(n)*a(n+1) = a(a(n)+n) = (a(n)+n)*(a(n+1)-n-1) + 1.
a(n) is never divisible by primes given in A003629.
Each odd prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -6 (mod p).
The prime factors are listed in A038873 and the primes in A028886.
For n > 0, this is a proper subsequence of A079896.
Conjecture: a(n) = A079896(A265284(n-1)). -
(End)

			From _Stefano Spezia_, Nov 08 2022: (Start)
Illustrations for n = 0..4:
          *       * * *     * * * * *
      a(0) = 1    *   *     *       *
                  * * *     *   *   *
                a(1) = 8    *       *
                            * * * * *
                            a(2) = 17
.
   * * * * * * *    * * * * * * * * *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   * * * * * * *    *   *   *   *   *
     a(3) = 28      *               *
                    * * * * * * * * *
                        a(4) = 41
(End)

Altug Alkan, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005563, A014616, A028560, A082111, A190576.

a014616 n = (n * (n + 6) + 1) `div` 4 -- Reinhard Zumkeller, Apr 07 2013

Range[3, 50]^2 - 8 (* Alonso del Arte, Aug 15 2016 *)

a(n)=(n+3)^2-8 \\ Charles R Greathouse IV, Oct 07 2015

(3 to 49).map(n => n * n - 8) // Alonso del Arte, May 07 2020

a(n) = a(n-1) + 2*n + 5 (with a(0) = 1). - Vincenzo Librandi, Aug 05 2010
a(n) = A028560(n) + 1; A014616(n) = floor(a(n+1)/4). - Reinhard Zumkeller, Apr 07 2013
G.f.: (-1 - 5*x + 4*x^2)/(x - 1)^3. - R. J. Mathar, Mar 24 2013
Sum_{n >= 0} 1/a(n) = 51/112 - Pi*cot(2*Pi*sqrt(2))/(4*sqrt(2)) = 1.3839174974448... . - Vaclav Kotesovec, Apr 10 2016
E.g.f.: (1 + 7*x + x^2)*exp(x). - G. C. Greubel, Aug 19 2017
Sum_{n >= 0} (-1)^n/a(n) = (-19 + 14*sqrt(2)*Pi*cosec(2*sqrt(2)*Pi))/112. - Amiram Eldar, Nov 04 2020
From Klaus Purath, Jan 04 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2, n >= 2.
a(n) = A082111(n) + n.
a(n) = A190576(n+1) - n. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = 7*Pi/(45*sqrt(2)*sin(2*sqrt(2)*Pi)).
Product_{n>=0} (1 + 1/a(n)) = (4*sqrt(14)/9)*sin(sqrt(7)*Pi)/sin(2*sqrt(2)*Pi). (End)

Definition corrected by Omar E. Pol, Jul 27 2009

A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).

Original entry on oeis.org

5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 245, 249, 253, 257

Offset: 1

Wolfdieter Lang, Nov 29 2002

The Pell equation x^2 - a(n)*y^2 = +4 has infinitely many (integer) solutions (see A077428 and A078355).
These are the odd numbers in A079896. The even ones are 4*A000037. - Wolfdieter Lang, Sep 15 2015
First differences: 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 8, ... , only 4's and 8's?. - Paul Curtz, Apr 11 2019
Yes. There are only 4's and 8's. Proof: Only multiples of 4 may appear. The 4's correspond to successive composite in A016813, whereas an 8 corresponds to a square. A greater multiple of 4 would imply to have at least 2 consecutive squares in A016813, which is not possible since 2 consecutive squares cannot have a difference of 4. That sequence of 4's and 8's can be obtained with A010052 (without the 1st term) where the 0's are replaced with 4's and 1's replaced with 8's. - Michel Marcus, Apr 16 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. R. Finch, Class number theory [Cached copy, with permission of the author]
A. M. Legendre, Expression les plus simples des formules Ly^2+Myz+Nz^2 où M est impair pour toutes les valeurs de B = M^2-4LN depuis B=5 jusqu'à B=305, Essai sur la Théorie des Nombres An VI, Table II. [_Paul Curtz_, Apr 11 2019]

Intersection of A016813 and A000037.

Cf. A077426, A079896.

A077425 := proc(n::integer) local resul,i ; resul := 5 ; i := 1 ; while i < n do resul := resul+4 ; while issqr(resul) do resul := resul+4 ; od ; i:= i+1 ; od ; RETURN(resul) ; end proc:
seq(A077425(n),n=1..31) ; # R. J. Mathar, Apr 25 2006

Select[Range[5,300,4],!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Dec 05 2012 *)

[n | n <- vector(100,n,4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014

list(lim)=my(v=List()); for(s=2,sqrtint((lim\=1)+1), forstep(n=s^2 + if(s%2,4,1), min((s+1)^2-1,lim), 4, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Nov 04 2021

from operator import sub
from sympy import integer_nthroot
def A077425(n): return n+sub(*integer_nthroot(n,2))<<2|1 # Chai Wah Wu, Oct 01 2024

More terms from Max Alekseyev, Mar 03 2010

A039955 Squarefree numbers congruent to 1 (mod 4).

Original entry on oeis.org

1, 5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 129, 133, 137, 141, 145, 149, 157, 161, 165, 173, 177, 181, 185, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 249, 253, 257, 265, 269

Offset: 1

R. K. Guy

The subsequence of primes is A002144.
The subsequence of semiprimes (intersection with A001358) begins: 21, 33, 57, 65, 69, 77, 85, 93, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265.
The subsequence with more than two prime factors (intersection with A033942) begins: 105 = 3 * 5 * 7, 165 = 3 * 5 * 11, 273, 285, 345, 357, 385, 429, 465. - Jonathan Vos Post, Feb 19 2011
Except for a(1) = 1 these are the squarefree members of A079896 (i.e., squarefree determinants D of indefinite binary quadratic forms). - Wolfdieter Lang, Jun 01 2013
The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la formule t^2+a*u^2, a étant de la forme 4 n + 1, Essai sur la Théorie des Nombres An VI, Table IV. See first column. [_Paul Curtz_, Aug 14 2019]

Cf. A002144, A039956, A039957, A005117, A185197.

a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+1: n in [0..67] | IsSquarefree(4*n+1)];  // Bruno Berselli, Mar 03 2011

fQ[n_] := Max[Last /@ FactorInteger@ n] == 1 && Mod[n, 4] == 1; Select[ Range@ 272, fQ] (* Robert G. Wilson v *)
Select[Range[1,300,4],SquareFreeQ[#]&] (* Harvey P. Dale, Mar 27 2020 *)

list(lim)=my(v=List([1])); forfactored(n=5,lim\1, if(vecmax(n[2][,2])==1 && n[1]%4==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

is(n)=n%4==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

A305312 Discriminant a(n) of the indefinite binary quadratic Markoff form m(n)*F_{m(n)}(x, y) with m(n) = A002559(n), for n >= 1.

Original entry on oeis.org

5, 32, 221, 1517, 7565, 10400, 71285, 257045, 338720, 488597, 1687397, 3348896, 8732021, 15800621, 22953677, 75533477, 157326845, 296631725, 376282400, 514518485, 741527357, 1078334240, 1945074605, 7391012837, 10076746685, 12768548000, 16843627085, 24001135925, 34830756896, 50658755621, 83909288237, 164358078917, 342312755621, 347220276512, 781553243021, 1636268213885, 2244540316037, 2379883179965, 3756053306912, 7713367517021

Offset: 1

Wolfdieter Lang, Jun 26 2018

Subsequence of A079896.
For the Markoff form f_{m(n)}(x, y) =  m(n)*F_{m(n)}(x, y) of Cassels (pp. 31-39), see the comments on A305310. Some references are given in A002559, A305308 and  A305310.
f_m(x, y) is an indefinite binary quadratic form because the discriminant is positive.
a(n) is also the discriminant D(n) = a(n) of the indefinite binary quadratic form determining the Markoff triple MT(n) = (x(n), y(n), m(n)) if the largest member is m(n) = A002559(n) and x(n) <= y(n) <= m(n). This is the form x^2 - 3*m*x*y + y^2 = -m^2 (with dropped argument n), or in reduced version  X^2 + b*X*Y - b*Y^2 = -m^2, with b = b(n) = 3*m(n) - 2, where X = X(n) = y(n) - x(n) and  Y = Y(n) = y(n). The uniqueness of such Markoff triples MT(n) with given largest members m(n) is a conjecture.
To find reduced forms one needs f(n) := ceiling(sqrt(D(n))) which is  3*m(n) because (3*m-1)^2 < 9*m^2 - 4 < (3*m)^2, due to 6*m(n) > 5, for n >= 1.
If the forms for a Markoff triple with largest member m are numerated with n giving m as m(n) = A002559(n)as in the present entry then the uniqueness conjecture is assumed to be true. Otherwise certain m(n) will lead to several different forms. - Wolfdieter Lang, Jul 30 2018

			a(5) = 7565 because 9*29^2 - 4 = 7565.

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.

Cf. A002559, A079896, A305308, A305310.

a(n) = 9*m(n)^2 - 4 = 9*A002559(n)^2 - 4, n >= 1.

A307359 Class number a(n) of indefinite binary quadratic forms with discriminant 4*A000037(n) for n >= 1.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 2, 4, 4, 3, 2, 4, 4, 1, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 1, 2, 4, 4, 2, 2, 2, 4, 1, 2, 4, 2, 4, 2, 2, 2, 4, 2, 4, 1, 2, 4, 2, 2, 4, 6, 4, 4, 2, 4, 2, 2, 4, 4, 1, 4, 4, 2, 2, 2, 4, 4, 1, 2, 8, 3, 4, 2, 4, 4, 2

Offset: 1

Wolfdieter Lang, Apr 04 2019

nonn

This is a subsequence of A087048, See the formula.
This sequence is relevant for the Pell forms [1, 0, - D(n)], with D(n) = A000037(n) and discriminant 4*D(n).
The Buell reference, Table 2B, pp. 241-243, gives only the class numbers, called there H, for A000037(n) squarefree and not congruent to 1 modulo 4. E.g., a(3), related to discriminant 4*5 = 20, is not treated there; also a(6) for discriminant 32 = 4*(2*2^2) does not appear there.
For the a(n) cycles of primitive reduced forms of discriminant 4*A000037(n) see the W. lang link in A324251, Table 2 and Table 1, for n = 1..30. - Wolfdieter Lang, Apr 19 2019

			a(1) = 1 because 4*A000037(1) = 4*2 = 8 = A079896(e(1)) with e(1) = 1 and A087048(1) = 1.
a(12) = 4 because the twelfth even number of A079896 is 60 at position e(12) = 22, and A087048(22) = 4.
The cycle for discriminant 8 is [[1, 2, -1], [-1, 2, 1]].
The four 2-cycles for discriminant 60 are  [[1, 6, -6], [-6, 6, 1]], [[-1, 6, 6], [6, 6, -1]], [[2, 6, -3], [-3, 6, 2]] and  [[-2, 6, 3], [3, 6, -2]].

D. A. Buell, Binary Quadratic Forms, Springer, 1989.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A000037, A079896, A082174, A087048.

def a(n):
    i, D, S = 1, 4*n + 4*floor(1/2 + sqrt(n)), []
    for b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            if gcd([a, b, (D-b^2)/(4*a)]) > 1: continue
            Q = BinaryQF(a, b, -(D-b^2)/(4*a))
            if all([(not Q.is_equivalent(t)) for t in S]): S.append(Q)
    return len(S)  # Robin Visser, Jun 01 2025

a(n) gives the number of distinct cycles of primitive reduced forms of discriminant 4*A000037(n).

a(n) = A087048(e(n)), with e(n) the position of the n-th even term of A079896, for n >= 1.

a(40) corrected and more terms from Robin Visser, Jun 01 2025

cp's OEIS Frontend

A261249 Number of classes of proper solutions of the Pell equation x^2 - D(n) y^2 = +4 for D(n) = A079896(n), n >= 1.

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A306638 a(n) is the norm of the fundamental unit of binary quadratic forms with discriminant D = A079896(n).

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Julia

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A257161 The length of the period under Zagier-reduction of the principal indefinite quadratic binary form of discriminant D(n) = A079896(n).

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A000037 Numbers that are not squares (or, the nonsquares).

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A014601 Numbers congruent to 0 or 3 mod 4.

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A028884 a(n) = (n + 3)^2 - 8.

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