A105750 -id:A105750 - cp's OEIS frontend

A363415 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-5).

1, 1, -9, -54, 426, 6426, -50274, -1465884, 10992996, 552727476, -3792193524, -312571718424, 1853425616616, 248005863100296, -1173524207653224, -263102748395914224, 865735128320476176, 359884863190774985616, -584551982838131141904, -616984573598760535235424, -155177934223071790979424

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes. Numerical calculation suggests that a similar division also holds in this case.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {5, 11, 13, 31, 53, 79, 97, 113, 137, 157, 179, 193, 197, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes begins {2, 3, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, ...} and consists of primes p == 1, 3, 7 or 9 (mod 20), equivalently, rational primes that split in the field extension Q(sqrt(-5)) of Q, together with the prime p = 2. See A139513.
It can be shown that the 2-adic valuation v_2(a(n)) = floor((n+1)/4).
Moll's conjecture 5.5 extends to this sequence: for type two primes p > 2, v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes begins {17, 19, 37, 59, 71, 73, 131, 151, 173, 191, 199, ...}.
Taken together, the type 1 and type 3 primes appear to consist of primes p == 11, 13, 17 or 19 (mod 20), equivalently, primes that remain inert in the field extension Q(sqrt(-5)) of Q, together with the prime p = 5, which ramifies in Q(sqrt(-5)). See A003626.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 0..80] = [0, 0, 2, 3, 1, 3, 3, 3, 4, 4, 4, 5, 5, 5, 7, 7, 7, 8, 8, 8, 9, 9, 9, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 16, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22, 22, 25, 25, 25, 26, 26, 26, 27, 27, 27, 29, 29, 29, 30, 30, 30, 31, 31, 31, 33, 33, 33, 34, 34, 34, 35, 35, 35, 39, 39, 40, 40].
Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 17: the sequence of 17-adic valuations [v_17(a(n)) : n = 0..100] = [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (5*n + 5)*tan(Sum_{k = 1..n} arctan(sqrt(5)*k))/sqrt(5).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
a(0) = a(1) = 1.

A236988 Real part of the product of all the Gaussian integers in the rectangle [1, 1] to [2, n].

Original entry on oeis.org

1, -20, 140, 200, -67600, 3983200, -228488000, 14375920000, -1002261520000, 74864404160000, -5398716356800000, 221997813232000000, 54286859023072000000, -27326116497867200000000, 9481971502321385600000000, -3155347494162485190400000000

Offset: 1

Jon Perry, Feb 02 2014

sign

By Gaussian integers, we mean complex numbers of the form a + bi, where both a and b are integers in Z, i = sqrt(-1). Thus the quadratic integer ring under consideration here is Z[i].

			For n = 2, we have (1 + i)(1 + 2i)(2 + i)(2 + 2i) which gives -20 + 0i, so a(2) = -20.

Cf. A105750, A234459, A204041.

function cNumber(x, y) {
return [x, y];
}
function cMult(a, b) {
return [a[0] * b[0] - a[1] * b[1], a[0] * b[1] + a[1] * b[0]];
}
for (i = 1; i < 20; i++) {
c = cNumber(1, 0);
for (j = 1; j <= 2; j++)
for (k = 1; k <= i; k++)
c = cMult(c, cNumber(j, k));
document.write(c[0] + ", ");
}

Table[Re[Times@@Flatten[Table[a + b I, {a, 2}, {b, n}]]], {n, 20}] (* Alonso del Arte, Feb 02 2014 *)

a(n) = real(prod(i=1, 2, prod(j=1, n, i+I*j))); \\ Michel Marcus, Feb 03 2014

a(n) +(2*n+3)*(n-2)*a(n-1) +n*(n+1)*(n^2-4*n+8)*a(n-2) -2*(n^2-4*n+8)*(n^2-4*n+5)*a(n-3)=0. - R. J. Mathar, Feb 08 2014

A363411 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-3).

Original entry on oeis.org

1, 1, -5, -32, 112, 2212, -5348, -292880, 276976, 64180144, 60400144, -21123205376, -68151050240, 9766562233792, 57568265355328, -6044149831446272, -54001800190537472, 4827069458763086080, 59568915131392086784, -4835221290238425841664, -77896195282519949963264

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes. Numerical calculation suggests that a similar division also holds in this case.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {3, 11, 17, 23, 29, 41, 47, 53, 59, 83, 89, 101, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 6), i.e., rational primes that split in the field extension Q(sqrt(-3)) of Q, together with the prime p = 2. See A002476.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.
(ii) for prime p == 1 (mod 6), the p-adic valuation v_p(a(n)) ~ n/(p - 1)
as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes begins {5, 71, 179, 191, 227, 257, 263, ...}.
Taken together, the type 1 and type 3 primes appear to consist of all primes p == 5 (mod 6), that is, the rational primes that remain inert in the field extension Q(sqrt(-3)) of Q, together with the prime p = 3, which ramifies in Q(sqrt(-3)). See A007528.

			Type 2 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..60] = [0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 6, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 9, 8, 8, 9, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10].
Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Table[Re[Product[1 + k*Sqrt[-3], {k, 0, n}]], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (3*n + 3)*tan(Sum_{k = 1..n} arctan(sqrt(3)*k))/sqrt(3).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) with
a(0) = a(1) = 1.
Conjecture: the 5-adic valuation v_5(a(n+2)) = A079998(n) (checked up to n =
5000).

A363412 a(n) = 1/sqrt(3) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-3).

Original entry on oeis.org

0, 1, 3, -12, -140, 420, 13692, -23744, -2366784, 126000, 641927440, 1306329024, -252172135488, -1138135788608, 135593735484480, 999117715814400, -95707279587325952, -1013737882826462976, 85873512374909086464, 1217682899871358735360, -95486742904897158097920

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes of the form p == 1 (mod 6), i.e., rational primes that split in the field extension Q(sqrt(-3)) of Q, together with the prime p = 2. See A002476.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.
(ii) for prime p == 1 (mod 6), the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes of the form p == 5 (mod 6), i.e., the rational primes that remain inert in the field extension Q(sqrt(-3)) of Q, together with the prime p = 3, which ramifies in Q(sqrt(-3)). See A007528.

			Type 2 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..60] = [0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 11, 10].
Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 0, 1, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Table[Im[Product[1 + k*Sqrt[-3], {k, 0, n}]] / Sqrt[3], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n-2*k).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) with a(0) = 0 and a(1) = 1.
Conjectures: 3 does not divide a(3*n+1) for all n; the 3-adic valuation v_3(a(3*n)) = v_3(a(3*n-1)) for all n.

A363413 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-4).

Original entry on oeis.org

1, 1, -7, -43, 245, 4045, -20795, -729335, 3118985, 217496825, -667140175, -97338843875, 149451128125, 61156245509125, 18055448952125, -51399370203595375, -123577855227019375, 55722247285947360625, 266112415762709595625, -75739843360243364046875, -560236984557463079546875

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105750(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes. Numerical calculation suggests that a similar division also holds in this case.
Type 1: primes that do not divide any element of the sequence {a(n)}.
We conjecture that the set of type 1 primes begins {2, 3, 11, 19, 23, 31, 47, 59, 67, 71, 79, ...}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q. See A002144.
Moll's conjecture 5.5 extends to this sequence and takes the form: for primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes begins {7, 43, 83, 131, 163, 167, ...}.
Taken together, the type 1 and type 3 primes appear to consist of primes p == 3 (mod 4), equivalently, primes that remain inert in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002145.

			Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 5, 3, 3, 3, 4, 4, 4, 6, 6, 5, 5, 5, 6, 6, 6, 6, 6, 8, 8, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 12, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14, 15, 15, 15, 15, 15, 17, 19, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 21, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 0..100] = [0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor((n+1)/2)} (-4)^k*Stirling1(n+1,n+1-2*k).
a(n+1)/a(n) = 1 - (2*n + 2)*tan( Sum_{k = 1..n} arctan(2*k) ).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
a(0) = a(1) = 1.

A363414 a(n) = (1/2) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-4).

Original entry on oeis.org

0, 1, 3, -18, -190, 1035, 25305, -120260, -5954940, 22115925, 2197084175, -5141457750, -1173207584250, 769657081375, 856957094209125, 1127788828491000, -821262134429035000, -2922085673288364375, 1000078365473764126875, 6056214264965246443750, -1508740652939902034493750

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of  Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
In this case, unlike in A105750, the set of type 1 primes is conjecturally empty; it appears that every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q. See A002144.
Moll's conjecture 5.5 extends to this sequence: for the primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002145.

			Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 1..100] = [0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 14, 12, 13, 12, 12, 14, 13, 14, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 19, 18, 19, 18, 18, 21, 19, 20, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..100] = [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor(n/2)} (-4)^k*Stirling1(n+1,n-2*k).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
a(0) = 0 and a(1) = 1.

cp's OEIS Frontend

A363415 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A236988 Real part of the product of all the Gaussian integers in the rectangle [1, 1] to [2, n].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

JavaScript

Mathematica

PARI

Formula

A363411 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A363412 a(n) = 1/sqrt(3) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A363413 a(n) = the real part of Product_{k = 0..n} 1 + k*sqrt(-4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A363414 a(n) = (1/2) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula