A257009 -id:A257009 - cp's OEIS frontend

A257007 Number of Zagier-reduced binary quadratic forms of discriminant n^2-4.

0, 0, 1, 3, 4, 7, 7, 12, 8, 20, 13, 18, 18, 31, 20, 31, 24, 39, 26, 53, 20, 66, 36, 36, 50, 76, 39, 62, 56, 92, 42, 72, 42, 120, 68, 72, 70, 136, 46, 126, 76, 112, 100, 96, 68, 146, 105, 125, 66, 226, 77, 168, 96, 138, 126, 160, 96, 228, 100, 142

Offset: 1

Barry R. Smith, Apr 16 2015

nonn

The number of finite sequences of positive integers with odd length parity and alternant equal to n.
The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=n^2-4.
The number of possible asymmetry types for the quotient sequence of the even-length continued fraction expansion of a rational number a/b, where b satisfies one of the congruences b^2 + nb + 1 = 0 (mod a) or b^2 - nb + 1 = 0 (mod a)

			For n=5, the a(5) = 4 Zagier-reduced forms of discriminant 21 are x^2 + 5*x*y + y^2, 5*x^2 + 9*x*y + 3*y^2, 3*x^2 + 9*x*y + 5*y^2, and 5*x^2 + 11*x*y + 5*y^2.

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

P. Kleban and A. Özlük, A Farey fraction spin chain, arXiv:cond-mat/9808182 [cond-mat.stat-mech], 1998; Communications in mathematical physics, 203(3):635-647, 1999. This sequence appears to be the function Phi(n) given in Theorem 4.
B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.
B. R. Smith, End-symmetric continued fractions and quadratic congruences, Acta Arith., 167 (2015) 173-187.
Marc Technau, The Calkin-Wilf tree and a trace condition, Master's Thesis, 2015. The sequence appears to be the function N(n,0) from subsection 1.3.1.
Marc Technau, Remark on the Farey fraction spin chain, arXiv:2304.08143 [math.NT], 2023. See Theorem 2 p. 4.

Cf. A257003, A257008, A257009.

It appears that this sequence gives half the row sums of the triangle in A264597 (cf. A264598), and also the first column of A264597. - N. J. A. Sloane, Nov 19 2015

# Maple code for the formula given by Kleban et al., which is almost certainly the same sequence as this (but until that is proved, the program should not be used to extend this sequence, A264598 or A264599). - N. J. A. Sloane, Nov 19 2015
with(numtheory); # return number of divisors of m less than b
dbm:=proc(b,m) local i,t1,t2;
t1:=divisors(m); t2:=0;
for i from 1 to nops(t1) do if t1[i]add(dbm(b,b*n-b^2-1), b=1..n-1);
[seq(f(n),n=1..100)];

Mathematica

Table[Length[ Flatten[ Select[ Table[{a, k}, {k, Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]], Mod[# - n, 2] == 0 &]}, {a, Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}], UnsameQ[#, {}] &], 1]], {n, Map[#^2 - 4 &, Range[3, 60]]}]

PARI

d(n, k) = #select(x->(xMichel Marcus, Apr 18 2023; based on Technau Lemma 3

Formula

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

From the Kleban et al. reference it appears that a(n) = Sum_{b=1..n-1} dbm(b,n*b-b^2-1), where dbm(b,m) = number of positive divisors of m that are less than b. - N. J. A. Sloane, Nov 19 2015

A257010 Number of sequences of positive integers with length 3 and alternant equal to n.

Original entry on oeis.org

0, 2, 2, 4, 3, 6, 2, 9, 4, 6, 5, 11, 4, 9, 6, 10, 5, 14, 2, 16, 7, 6, 9, 16, 6, 11, 8, 17, 5, 14, 4, 20, 10, 8, 9, 22, 2, 17, 10, 16, 11, 14, 6, 18, 13, 12, 5, 28, 6, 19, 9, 15, 13, 16, 8, 24, 6, 12, 11, 32, 6, 15, 16, 16, 9, 19, 8, 30, 8, 14, 9, 30, 8, 15, 12, 21, 16, 22

Offset: 3

Barry R. Smith, Apr 18 2015

nonn

See A257009 for the definition of the alternant of a sequence. The number of sequences of length 1 with given alternant value n is 1, while the number of sequences of length 2 with given alternant value n is d(n), the number of divisors of n (see A000005).

There are infinitely many sequences of length 3 and alternant equal to 2. It is for this reason that the offset is 3.

			For n=6, the a(6) = 4 sequences with alternant 6 are (1,1,3), (1,3,2), (2,3,1), (3,1,1)

Andrew Howroyd, Table of n, a(n) for n = 3..1000
B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.

Cf. A257009, A257011, A257012, A257013, A000012, A000005.

Dbm:= proc(b,m) nops(select(t -> (t-1) mod b = 0, numtheory:-divisors(m))) end proc:
seq(add(Dbm(b,b^2+n*b+1)-2, b=1..n-1), n=3..100); # Robert Israel, Jan 24 2016

Length3Q[x_, y_] :=
Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[OddQ[Length[l]], Return[Length[l] == 3],
   If[Last[l] == 1, Return[Length[l] - 1 == 3], Return[Length[l] + 1 == 3]]]];
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 - 4]], Floor[Sqrt[n^2 - 4]]],
        Mod[# - n^2 + 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 - 4 - k^2)/4], # > (Sqrt[n^2 - 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length3Q[#, n] &]], {n, 3, 60}]

a(n)={sum(b=1, n-1, sumdiv(b^2+n*b+1, d, (d-1)%b==0) - 2)} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{b=1..n-1} (Dbm (b,b^2+nb+1)-2), where Dbm(b,m) is the number of positive divisors of m that are congruent to 1 modulo b. - Barry R. Smith, Jan 24 2016

Terms a(61) and beyond from Andrew Howroyd, May 01 2020

A257011 Number of sequences of positive integers with length 4 and alternant equal to n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 5, 8, 11, 10, 10, 20, 11, 16, 21, 24, 15, 26, 23, 28, 31, 22, 24, 49, 27, 36, 33, 36, 33, 52, 33, 46, 51, 42, 41, 64, 41, 38, 54, 74, 43, 64, 44, 66, 63, 56, 57, 88, 59, 58, 79, 60, 52, 96, 61, 92, 69, 68, 72, 110

Offset: 1

Barry R. Smith, Apr 18 2015

nonn

			For n=5, the a(5)=3 sequences with alternant 5 and length 4 are (1,3,1,1), (1,2,2,1), and (1,1,3,1).

B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.

Cf. A257009, A257010, A257012, A257013, A000012, A000005

Length4Q[x_, y_] :=
Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[EvenQ[Length[l]], Return[Length[l] == 4],
   If[Last[l] == 1, Return[Length[l] - 1 == 4], Return[Length[l] + 1 == 4]]]];
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 + 4]], Floor[Sqrt[n^2 + 4]]],
        Mod[# - n^2 - 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 + 4 - k^2)/4], # > (Sqrt[n^2 + 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length4Q[#, n] &]], {n, 1, 60}]

A257012 Number of sequences of positive integers with length 5 and alternant equal to n.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 10, 8, 11, 11, 19, 15, 19, 17, 27, 17, 36, 17, 43, 27, 29, 31, 54, 30, 41, 45, 63, 29, 57, 33, 75, 49, 59, 47, 96, 39, 79, 57, 84, 61, 81, 49, 97, 81, 85, 47, 150, 64, 105, 75, 101, 69, 123, 77, 141, 81, 103, 71, 189, 75, 119, 121, 137, 82, 143, 85, 183, 101, 129, 93, 211, 89, 129, 131, 187, 116, 201

Offset: 1

Barry R. Smith, Apr 19 2015

nonn

			The a(7) = 3 sequences with length 5 and alternant 7 are (1,1,1,3,1), (1,2,1,2,1), and (1,3,1,1,1).

B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.

Cf. A257009, A257010, A257011, A257013, A000012, A000005

Length5Q[x_, y_] :=
 Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[OddQ[Length[l]], Return[Length[l] == 5],
   If[Last[l] == 1, Return[Length[l] - 1 == 5], Return[Length[l] + 1 == 5]]]];
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 - 4]], Floor[Sqrt[n^2 - 4]]],
        Mod[# - n^2 + 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 - 4 - k^2)/4], # > (Sqrt[n^2 - 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length5Q[#, n] &]], {n, 3, 80}]

A257013 Number of sequences of positive integers with length 6 and alternant equal to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 5, 0, 4, 4, 9, 0, 12, 1, 13, 10, 8, 4, 33, 4, 14, 12, 21, 4, 44, 2, 33, 22, 24, 12, 62, 8, 16, 29, 63, 2, 64, 4, 57, 52, 26, 10, 111, 21, 40, 48, 45, 8, 106, 26, 94, 40, 46, 18, 164, 21, 40, 61, 97, 40, 118, 12, 87, 65, 104, 14, 221, 14, 52, 116, 88, 30, 146, 21, 157

Offset: 1

Barry R. Smith, Apr 19 2015

nonn

			For n=14, the a(14)=4 sequences with alternant 14 and length 6 are (1,1,1,1,4,1), (1,2,1,1,3,1), (1,3,1,1,2,1), and (1,4,1,1,1,1).

B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.

Cf. A257009, A257010, A257011, A257012, A000012, A000005

Length6Q[x_, y_] :=
 Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[EvenQ[Length[l]], Return[Length[l] == 6],
   If[Last[l] == 1, Return[Length[l] - 1 == 6], Return[Length[l] + 1 == 6]]]]
Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n^2 + 4]], Floor[Sqrt[n^2 + 4]]],
        Mod[# - n^2 - 4, 2] == 0 &]}, {a,
       Select[Divisors[(n^2 + 4 - k^2)/4], # > (Sqrt[n^2 + 4] - k)/2 &]}],
     UnsameQ[#, {}] &], 1], Length6Q[#, n] &]], {n, 1, 80}]

A257008 Number of Zagier-reduced binary quadratic forms of discriminant n^2+4.

Original entry on oeis.org

1, 2, 3, 5, 5, 10, 7, 13, 14, 16, 12, 31, 13, 24, 29, 38, 17, 44, 26, 47, 46, 34, 30, 90, 34, 56, 49, 63, 39, 106, 40, 87, 77, 70, 57, 139, 55, 58, 89, 149, 52, 138, 52, 136, 123, 92, 69, 223, 84, 104, 146, 111, 62, 218, 94, 214, 121, 132, 96, 296

Offset: 1

Barry R. Smith, Apr 16 2015

nonn

The number of finite sequences of positive integers with even length parity and alternant equal to n.
The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=n^2+4.
The number of possible asymmetry types for the quotient sequence of the odd-length continued fraction expansion of a rational number a/b, where b satisfies one of the congruences b^2 + nb - 1 = 0 (mod a) or b^2 - nb - 1 = 0 (mod a)

			For n=4, the a(4) = 5 Zagier-reduced forms of discriminant 20 are x^2 + 6*x*y + 4*y^2, 4*x^2 + 6*x*y + y^2, 4*x^2 + 10*x*y + 5*y^2, 5*x^2 + 10*x*y + 4*y^2, and 2*x^2 + 6*x*y + 2*y^2

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

B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.
B. R. Smith, End-symmetric continued fractions and quadratic congruencesActa Arith., 167 (2015) 173-187.

Cf. A257003, A257007, A257009

Table[Length[
  Flatten[
   Select[
    Table[{a, k}, {k,
      Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
       Mod[# - n, 2] == 0 &]}, {a,
      Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
    UnsameQ[#, {}] &], 1]], {n, Map[#^2 + 4 &, Range[3, 60]]}]

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

cp's OEIS Frontend

A257007 Number of Zagier-reduced binary quadratic forms of discriminant n^2-4.

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A257010 Number of sequences of positive integers with length 3 and alternant equal to n.

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A257011 Number of sequences of positive integers with length 4 and alternant equal to n.

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A257012 Number of sequences of positive integers with length 5 and alternant equal to n.

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A257013 Number of sequences of positive integers with length 6 and alternant equal to n.

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A257008 Number of Zagier-reduced binary quadratic forms of discriminant n^2+4.

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Crossrefs

Programs

Mathematica

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