A000003 -id:A000003 - cp's OEIS frontend

A000490 Generalized Euler numbers c(4,n).

1, 16, 1280, 249856, 90767360, 52975108096, 45344872202240, 53515555843342336, 83285910482761809920, 165262072909347030040576, 407227428060372417275494400, 1219998300294918683087199010816, 4366953142363907901751614431559680, 18406538229888710811704852978971181056

Offset: 0

N. J. A. Sloane

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).

Matthew House, Table of n, a(n) for n = 0..194
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Row 4 of A235605.

Cf. A000187, A000436, A000318, A349264.

egf := sec(4*x): ser := series(egf, x, 26):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

a0 = 4; nmax = 20; km0 = nmax; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2*k+1]/(2*k+1)^s, {k, 0, km}]; c[a_, n_, km_] := 2^(2*n +1)*Pi^(-(2*n)-1)*(2*n)!*a^(2*n+1/2)*L[a, 2*n+1, km] // Round; cc[km_] := cc[km] = Table[c[a0, n, km], {n, 0, nmax}]; cc[km0]; cc[km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000490 = cc[km] (* Jean-François Alcover, Feb 05 2016 *)
Range[0, 26, 2]! CoefficientList[Series[Sec[4 x], {x, 0, 26}], x^2] (* Matthew House, Oct 05 2024 *)

a(n) = A000364(n)*16^n. - Philippe Deléham, Oct 27 2006

a(n) = (2*n)!*[x^(2*n)](sec(4*x)). - Peter Luschny, Nov 21 2021

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 02 2000

A001587 Generalized Euler numbers.

Original entry on oeis.org

2, 6, 46, 522, 7970, 152166, 3487246, 93241002, 2849229890, 97949265606, 3741386059246, 157201459863882, 7205584123783010, 357802951084619046, 19133892392367261646, 1096291279711115037162, 67000387673723462963330, 4350684698032741048452486, 299131045427247559446422446

Offset: 0

N. J. A. Sloane

nonn

These numbers are related to the values at negative integers of the L-functions for two primitive Dirichlet characters of conductor 24. - F. Chapoton, Oct 05 2020

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).

Lars Blomberg, Table of n, a(n) for n = 0..199
LMFDB, character 24.5
LMFDB, character 24.11
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigendum: Generalized Euler and class numbers, Math. Comp. 22, (1968) 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Bisections are A000192 and A000411. Overview in A349264.

Similar sequences: A000111, A225147.

egf := sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)): ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # Peter Luschny, Nov 21 2021

t = PowerSeriesRing(QQ, 't').gen()
f = 2 * (sin(3 * t) + cos(3 * t)) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list() # F. Chapoton, Oct 06 2020

E.g.f.: 2 (sin(3 x) + cos(3 x)) / (2 cos(4 x) - 1). - F. Chapoton, Oct 06 2020
a(n) ~ 2^(2*n + 2) * 3^(n + 1/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Nov 05 2021
a(n) = n!*[x^n](sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x))). - Peter Luschny, Nov 21 2021

a(11)-a(14) from Lars Blomberg, Sep 10 2015

A091967 a(n) is the n-th term of sequence A_n, ignoring the offset, or -1 if A_n has fewer than n terms.

Original entry on oeis.org

0, 2, 1, 0, 2, 3, 0, 6, 6, 4, 44, 1, 180, 42, 16, 1096, 7652, 13781, 8, 24000, 119779, 458561, 152116956851941670912, 1054535, -53, 26, 27, 59, 4806078, 2, 35792568, 3010349, 2387010102192469724605148123694256128, 2, 0, -53, 43, 0, -4097, 173, 37338, 111111111111111111111111111111111111111111, 30402457, 413927966

Offset: 1

Proposed by several people, including Jeff Burch and Michael Joseph Halm

sign

This version ignores the offset of A_n and just counts from the beginning of the terms shown in the OEIS entry.
Thus a(8) = 6 because A_8 begins 1,1,2,2,3,4,5,6,... [even though A_8(8) is really 7].
The value a(n) = -1 could arise in two different ways, but it will be easy to decide which. - N. J. A. Sloane, Nov 27 2016
From M. F. Hasler, Sep 22 2013: (Start)
The value of a(91967) can be chosen at will.
Note that this sequence may change if the initial terms in A_n are altered, which does happen from time to time, usually because of the addition of an initial term.
After a(47), currently unknown, the sequence continues with a(48) = A48(47) = 1497207322929, a(49) = A49(48) = unknown, a(50) = A50(49) = unknown, a(51) = A51(50) = 1125899906842625, a(52)=97, a(53) = -1 (since A000053 has only 29 terms). (End)
a(58) = A000058(57) = 138752...985443 (29334988649136302 digits) is too large to include in the b-file. - Pontus von Brömssen, May 21 2022

			a(1) = 0 since A000001 has offset 0, and begins with A000001(0) = 0.
a(26) = 26 because the 26th term of A000026 = 26.

Pontus von Brömssen, Table of n, a(n) for n = 1..57 (terms n = 1..46 from M. F. Hasler)
OEIS, List of finite sequences, sorted by A-number.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences whose definition involves A_n (or An).

See A051070, A107357, A102288 for other versions.

Cf. A000001, A000002, A000003, A000004, A000005, A000006, A000007, A000008, A000009, A000010, A000011, A000012, A000013, A000014, A000015, etc.

Corrected and extended by Jud McCranie; further extended by N. J. A. Sloane and E. M. Rains, Dec 08 1998
Corrected and extended by N. J. A. Sloane, May 25 2005
a(26), a(36) and a(42) corrected by M. F. Hasler, Jan 30 2009
a(43) and a(44) added by Daniel Sterman, Nov 27 2016
a(1) corrected by N. J. A. Sloane, Nov 27 2016 at the suggestion of Daniel Sterman
Definition and comments changed by N. J. A. Sloane, Nov 27 2016

A232550 Number of distinct primitive quadratic forms of discriminant = -4n that exist such that every prime p for which p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n) can be represented by one of them.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 5, 1, 2, 3, 2, 2, 2, 2, 2, 3, 2, 1, 4, 1, 2, 3, 2, 2, 2, 1, 3, 2, 2, 2, 5, 2, 1, 3, 2, 1, 4, 2, 2, 2, 1, 3, 3, 2, 2, 3, 2, 2

Offset: 1

V. Raman, Nov 26 2013

A quadratic form is primitive if the GCD of the coefficients is 1. For example, the quadratic form 2*x^2+4*y^2 is not primitive.
Two quadratic forms f(x,y) = a*x^2+b*x*y+c*y^2 and g(x,y) = p*x^2+q*x*y+r*y^2 are distinct (or inequivalent) if and only if one cannot be obtained by a linear transformation (of the variables x, y) from the other. For example, the three quadratic forms u(x,y) = 3*x^2+2*x*y+3*y^2, v(x,y) = 3*x^2+4*x*y+4*y^2 and w(x,y) = 3*x^2+10*x*y+11*y^2 are equivalent because v(x,y) = u(x+y,-y) and w(x,y) = v(x+y,y). Also, w(x,y) = u(x+2*y,-y). Similarly, the two quadratic forms s(x,y) = 8*x^2+9*y^2 and t(x,y) = 17*x^2+50*x*y+41*y^2 are equivalent because t(x,y) = s(x+2*y,x+y).
The quadratic form x^2+n*y^2 is one such form and the only such form if n is a convenient number (A000926).
a(n) = 1 if and only if n is a convenient number (A000926).
Any prime p such that p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n) can be represented by exactly one of the a(n) distinct primitive quadratic forms of discriminant = -4n in at most four different ways (if n >= 2) or in at most eight different ways (if n = 1).
If a prime p can be written in the form x^2+n*y^2, then either p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n), assuming that p^2 does not divide n.
For primes p such that p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n), there is a lowest square m^2 such that m^2*p can be written in form x^2+n*y^2, where x and y are nonnegative integers (see A232529 and A232530).
If n is a prime congruent to 3 (mod 4), then a(n) = A232551(n).
The product of two numbers (prime or composite, same or different) which can be represented by the same quadratic form of discriminant = -4n can be written in the form x^2+n*y^2, as the following identity shows.
(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*Z+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(a*d-b*c)^2.
(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(b*d*(Y/X)+a*d+b*c)^2.
Note that for the latter equation, (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2)) and (b*d*(Y/X)+a*d+b*c) need not always be integers. If they are both integers, then it will be a second representation of the product of (X*a^2+Y*a*b+Z*b^2) and (X*c^2+Y*c*d+Z*d^2) in the form x^2+((X*Z)-(Y^2/4))*y^2.

			If n is a convenient number (A000926), then the only such available quadratic form is x^2+n*y^2.
For n = 11, every prime that is congruent to {0, 1, 3, 4, 5, 9} mod 11 can be represented by either of the two distinct primitive quadratic forms of discriminant = -44: x^2+11*y^2 or 3*x^2+2*x*y+4*y^2.
For n = 14, every prime that is congruent to {1, 2, 7, 9, 15, 23, 25, 39} mod 56 can be represented by either of the two distinct primitive quadratic forms of discriminant = -56: x^2+14*y^2 or 2*x^2+7*y^2.
For n = 17, every prime that is congruent to {1, 2, 9, 13, 17, 21, 25, 33, 49, 53} mod 68 can be represented by either of the two distinct primitive quadratic forms of discriminant = -68: x^2+17*y^2 or 2*x^2+2*x*y+9*y^2.
For n = 19, every prime that is congruent to {0, 1, 4, 5, 6, 7, 9, 11, 16, 17} mod 19 can be represented by either of the two distinct primitive quadratic forms of discriminant = -76: x^2+19*y^2 or 4*x^2+2*x*y+5*y^2.
For n = 20, every prime that is congruent to {1, 5, 9} mod 20 can be represented by either of the two distinct primitive quadratic forms of discriminant = -80: x^2+20*y^2 or 4*x^2+5*y^2.

V. Raman, Illustration of above mentioned quadratic forms for n = 1..1000

Cf. A000003, A000926, A232529, A232530, A232551 (Number of distinct primitive quadratic forms of discriminant = -4*n needed to generate all primes p for which -n is a quadratic residue (mod p)).

A232551 Number of distinct primitive quadratic forms of discriminant -4n that exist such that every prime p for which -n is a quadratic residue mod p can be represented by one of them.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 4, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 3, 4, 3, 4, 3, 2, 4, 3, 4, 5, 4, 2, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 4, 3, 6, 4, 2, 5, 4, 4, 5, 3, 3, 6, 6, 2, 5, 6, 4, 4, 4, 3, 6, 4, 4, 6, 4, 3, 6, 4, 3, 5, 6, 4, 6, 4, 4, 7, 6, 4, 4, 4, 5, 5, 6, 3, 5, 4, 3

Offset: 1

V. Raman, Nov 26 2013

This is closely related to the class number problem.
A quadratic form is primitive if the GCD of the coefficients is 1. For example, the quadratic form 2*x^2+4*y^2 is not primitive.
Two quadratic forms f(x,y) = a*x^2+b*x*y+c*y^2 and g(x,y) = p*x^2+q*x*y+r*y^2 are distinct (or inequivalent) if and only if one cannot be obtained by a linear transformation (of the variables x, y) from the other. For example, the three quadratic forms u(x,y) = 3*x^2+2*x*y+3*y^2, v(x,y) = 3*x^2+4*x*y+4*y^2 and w(x,y) = 3*x^2+10*x*y+11*y^2 are equivalent because v(x,y) = u(x+y,-y) and w(x,y) = v(x+y,y). Also, w(x,y) = u(x+2*y,-y). Similarly, the two quadratic forms s(x,y) = 8*x^2+9*y^2 and t(x,y) = 17*x^2+50*x*y+41*y^2 are equivalent because t(x,y) = s(x+2*y,x+y).
The quadratic form x^2+n*y^2 is one such form and the only such form if n = 1, 2, 3, 4, 7.
a(n) = 1 if and only if n = 1, 2, 3, 4, 7.
If n is a squarefree convenient number (A000926), a(n) represents the class number of the ring Z[sqrt(n)] if n == 1 (mod 4) or if n == 2 (mod 4) and the class number of the ring Z[(1+sqrt(n))/2] if n == 3 (mod 4) and this class number is a power of 2.
Any prime p such that -n is a quadratic residue mod p can be represented by exactly one of the a(n) distinct primitive quadratic forms of discriminant = -4n in at most four different ways (if n >= 2) or in at most eight different ways (if n = 1).
If n is a prime congruent to 3 (mod 4), then a(n) = A232550(n).
If p is a prime, p^2 does not divide n, and p > 2 if n == 3 (mod 8), then there is a multiple of p in which p is raised to an odd power which can be written in the form x^2+n*y^2 if and only if -n is a quadratic residue mod p.
The product of two numbers (prime or composite, same or different) which can be represented by the same quadratic form of discriminant = -4n can be written in the form x^2+n*y^2, as the following identity shows:
(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*Z+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(a*d-b*c)^2.
(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(b*d*(Y/X)+a*d+b*c)^2.
Note that for the latter equation, (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2)) and (b*d*(Y/X)+a*d+b*c) need not always be integers. If they are both integers, then it will be a second representation of the product of (X*a^2+Y*a*b+Z*b^2) and (X*c^2+Y*c*d+Z*d^2) in the form x^2+((X*Z)-(Y^2/4))*y^2.
This sequence is the same as taking every fourth number in A107628. - T. D. Noe, Jan 02 2014

			If n = 1, 2, 3, 4 or 7, then the only such available quadratic form is x^2+n*y^2.
For n = 5, every prime that is congruent to {1, 2, 3, 5, 7, 9} mod 20 can be represented by either of the two distinct primitive quadratic forms of discriminant = -20: x^2+5*y^2 or 2*x^2+2*x*y+3*y^2.
For n = 6, every prime that is congruent to {1, 2, 3, 5, 7, 11} mod 24 can be represented by either of the two distinct primitive quadratic forms of discriminant = -24: x^2+6*y^2 or 2*x^2+3*y^2.
For n = 10, every prime that is congruent to {1, 2, 5, 7, 9, 11, 13, 19, 23, 37} mod 40 can be represented by either of the two distinct primitive quadratic forms of discriminant = -40: x^2+10*y^2 or 2*x^2+5*y^2.

V. Raman, Illustration of above mentioned quadratic forms for n = 1..1000

Cf. A000003, A000926, A232529, A232530, A232550 (Number of distinct primitive quadratic forms of discriminant = -4*n needed to generate all primes p for which p is a quadratic residue (mod 4*n) or p-n is a quadratic residue (mod 4*n)).

A244819 Positive numbers primitively represented by the binary quadratic form (1, 0, 3).

Original entry on oeis.org

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -12.

x^2 + 3*y^2 represents positive k properly (gcd(x, y) = 1), with nonnegative x, and the following multiplicities m(k): m(1) = 1, m(3) = 1, m(4) = 2, and if k = 3^a*4^b*Product_{j=1..P1} p1(j)^e1(j), with p1(j) primes 1 (mod 6) (A002476), e1(j) nonnegative integer numbers, and a and b from {0, 1}, then m(k) = 2^(P1+b). Shown by the lifting theorem (e.g., Apostol) for prime powers. Note that for prime 2 there is one solution of j^2 + 3 == 0 (mod 2) this corresponds the imprimitive reduced form (2, 2, 2), not to the one reduced primitive form (1, 0, 3) for discriminant -12 (A000003(3) = 1). - Wolfdieter Lang, Mar 02 2021

			Proper solution to x^2 + 3*y^2 = a(n), with x nonnegative: a(12 = 3*4) with (x, y) = (3, pm 1), pm = +1 or -1, multiplicity m(12) = 2, (a, b, P1) = (1, 1, 0); a(21 = 3*7) with (3, pm 2), m(21) = 2, (a, b, P1) = (1, 0, 1); a(49 = 7^2) with (1, pm 4), m(49) = 2 (a, b, P1) = (0, 0, 1)). - _Wolfdieter Lang_, Mar 02 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976 (1986), Theorem 5.30, pp. 121-122.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002476, A092574, A244779, A244780.

# Function PriRepBQF in A244779.
A244819 list := n -> PriRepBQF(1, 0, 3, n); A244819_list(259);

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + 3 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A276183 Genus of the quotient of the modular curve X_0(n) by the Fricke involution.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 3, 0, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 4, 0, 2, 2, 2, 1, 3, 2, 5, 1, 2, 1, 4, 1, 4, 3, 3, 2, 4, 1, 4, 2, 4, 4, 4, 1, 3, 3, 2, 3, 3, 1, 7

Offset: 1

Gheorghe Coserea, Oct 21 2016

nonn

a(n) is the genus of quotient space H/Gamma_0*(n), where H is the upper half plane and Gamma_0*(n) = Gamma_0(n) + W Gamma_0(n) is the extension of Gamma_0(n) via the involution z <-> W(z) = -n/z (see Cohn, 1988).

			G.f. = x^22 + x^28 + x^30 + x^33 + x^34 + x^37 + x^38 + x^40 + 2*x^42 + x^43 + x^44 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..54321
Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.
Harvey Cohn, A Numerical Survey of the Reduction of Modular Curve Genus by Fricke's Involutions, Number Theory (New York Seminar 1989-1990), p. 100.
Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Andrew P. Ogg, Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, vol. 16, no. 1 (1974-1975), talk no. 7, p. 1.

Cf. A000003, A000086, A000089, A001615, A001616, A001617, A276181.

f[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@ n}] - Count[(#^2 - # + 1)/n & /@ Range@ n, ?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range@ n, ?IntegerQ]/4];
g[n_] := Ceiling[k0 = k /. FindRoot[EllipticK[1 - k^2]/EllipticK[k^2] == Sqrt@ n, {k, 1/2, 10^-10, 1}, WorkingPrecision -> 600, MaxIterations -> 100]; Exponent[MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2];
r[n_] := If[MemberQ[{3, 7}, #], 3 + (# - 1)/2, 3] &@ Mod[n, 8]; a[n_] := If[n <= 4, 0, (1 + f@ n)/2 - r[n] g[n]/12]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 102}] (* Michael De Vlieger, Oct 28 2016, after Michael Somos at A001617 and Jean-François Alcover at A000003 *)
ClassList[n_?Negative] :=
Select[Flatten[#, 1] &@Table[
    {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}],
  Mod[#3, 1] == 0 && #3 >= # &&
      GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]
A001617[n_] := If[n < 1, 0,
  1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d,
     Divisors@n}] -
   Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 -
   Count[(#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4];
a[n_] := If[0 <= n <= 4, 0, (A001617[n] + 1)/2 - If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]] Length[ClassList[-4 n]]/12] (* David Jao, Sep 07 2020 *)

A000003(n) = qfbclassno(-4*n);
A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k, 1]+1)),
     h = prod(k=1, fsz, f[k, 1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
a(n) = {
  my(r = if (n%8 == 3, 4, n%8 == 7, 6, 3));
  if (n < 5, 0, (1 + A001617(n))/2 - r * A000003(n)/12);
};
vector(102, n, a(n))

a(n) = (1 + A001617(n))/2 - r * A000003(n)/12 for all n > 4, where r=4 for n=3 (mod 8), r=6 for n=7 (mod 8) and r=3 otherwise.

a(n) <> 4884 for all n.

New name from David Jao, Sep 07 2020

A000362 Generalized class numbers c_(n,2).

Original entry on oeis.org

5, 57, 352, 1280, 3522, 7970, 15872, 29184, 49410, 79042, 122400, 180224, 257314, 362340, 492032, 655360, 867588, 1117314, 1420320, 1803264, 2237380, 2745154, 3380736, 4080640, 4881250, 5874150, 6928416, 8126464, 9600870, 11133604

Offset: 1

N. J. A. Sloane

Let L_a(s) = Sum_{k>=0} (-a|2k+1) /(2k+1)^s be a Dirichlet series, where (-a|2k+1) is the Jacobi symbol. Then the c_(a,n) are defined by L_a(2n+1) = (Pi/(2a))^(2n+1)*sqrt(a)*c_(a,n)/(2n)! for n=0,1,2,..., a=1,2,3,...

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).

Matthew House, Table of n, a(n) for n = 1..10000
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Cf. A000233, A000508.

amax = 30;   km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k+1]/(2k+1)^s, {k, 0, km}]; c[1, n_, km_] := 2(2n)! L[1, 2n+1, km] (2 / Pi)^(2n+1) // Round; c[a_ /; a>1, n_, km_] := (2n)! L[a, 2n+1, km] (2a / Pi)^(2n+1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[c[a, n, km], {a, 1, amax}, {n, 0, nmax}]; cc[km0]; cc[km = 2km0]; While[cc[km] != cc[km/2, km = 2km]]; A000362[a_] := cc[km][[a, 3]]; Table[A000362[a], {a, 1, amax} ] (* Jean-François Alcover, Feb 08 2016 *)
Table[rowA235605[n, 2][[3]], {n, 50}] (* see A235605 *) (* Matthew House, Oct 05 2024 *)

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 02 2000

A000508 Generalized class numbers c_(n,3).

Original entry on oeis.org

61, 2763, 38528, 249856, 1066590, 3487246, 9493504, 22634496, 48649086, 96448478, 179369856, 315621376, 530788622, 860061996, 1346126848, 2046820352, 3038120316, 4403100222, 6254596992, 8737505280, 11992903772

Offset: 1

N. J. A. Sloane

Let L_a(s) = Sum_{k>=0} (-a|2k+1) /(2k+1)^s be a Dirichlet series, where (-a|2k+1) is the Jacobi symbol. Then the c_(a,n) are defined by L_a(2n+1) = (Pi/(2a))^(2n+1)*sqrt(a)*c_(a,n)/(2n)! for n=0,1,2,..., a=1,2,3,...

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).

Matthew House, Table of n, a(n) for n = 1..10000
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 1967 6890694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Column 3 of A235605.

Cf. A000003, A000233, A000362.

amax = 25; km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[ -a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; c[1, n_, km_] := 2 (2 n)! L[1, 2 n + 1, km] (2/Pi)^(2 n + 1) // Round; c[a_ /; a > 1, n_, km_] := (2 n)! L[a, 2 n + 1, km] (2 a/Pi)^(2 n + 1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[ c[a, 3, km], {a, 1, amax} ]; cc[km0]; cc[km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000508 = cc[km] (* Jean-François Alcover, Feb 09 2016 *)
Table[rowA235605[n, 3][[4]], {n, 50}] (* see A235605 *) (* Matthew House, Oct 05 2024 *)

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 02 2000

Name clarified by James C. McMahon, Nov 30 2023

A067752 Number of unordered solutions of xy + xz + yz = n in nonnegative integers.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 4, 2, 3, 4, 4, 3, 3, 3, 5, 4, 2, 4, 6, 3, 4, 5, 4, 4, 4, 4, 6, 4, 3, 6, 7, 2, 4, 6, 6, 5, 4, 3, 7, 6, 3, 6, 8, 4, 5, 6, 5, 4, 6, 6, 9, 4, 2, 7, 8, 4, 5, 8, 7, 6, 6, 3, 8, 6, 4, 8, 9, 3, 6, 8, 7, 6, 4, 6, 11, 7, 3, 7, 10, 4, 6, 8, 6, 7

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe, Jun 14 2006

a(n) is also the total number of distinct quadratic forms of discriminant -4n. A232551 counts only the primitive quadratic forms of discriminant -4n (those with all coefficients pairwise coprime) and A234287 includes those by which some prime can be represented (those with all coefficients pairwise coprime or coefficient of x^2 is prime or coefficient of y^2 is prime). This sequence includes all quadratic forms like 2x^2 + 2xy + 4y^2 and 2x^2 + 4y^2 which are non-primitive and those like 4x^2 + 2xy + 4y^2 and 4x^2 + 4xy + 4y^2 by which no prime can be represented (those with no restrictions). - V. Raman, Dec 24 2013

			a(12)=4 because of (0,1,12), (0,2,6), (0,3,4), (2,2,2).
a(20)=5 because of (0,1,20), (0,2,10), (0,4,5), (1,2,6), (2,2,4).

T. D. Noe, Table of n, a(n) for n = 1..10000
V. Raman, Examples of these distinct quadratic forms for n = 1..100

Cf. A000003, A000926, A067751, A067753, A067754, A232550, A232551, A234287.

Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt++ ], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected, extended and edited by John W. Layman, Dec 03 2004

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