A234459 -id:A234459 - cp's OEIS frontend

A219086 a(n) = floor((n + 1/2)^4).

0, 5, 39, 150, 410, 915, 1785, 3164, 5220, 8145, 12155, 17490, 24414, 33215, 44205, 57720, 74120, 93789, 117135, 144590, 176610, 213675, 256289, 304980, 360300, 422825, 493155, 571914, 659750, 757335, 865365, 984560, 1115664

Offset: 0

Clark Kimberling, Jan 01 2013

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/4 and { } = fractional part.  Equivalently, the jump sequence of f(x) = x^(1/4), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p).  For details and a guide to related sequences, see A219085.
-4*a(n) gives the real part of (n+n*i)*((n+1)+n*i)*(n+(n+1)*i)*((n+1)+(n+1)*i). The imaginary part is always zero. - Jon Perry, Feb 05 2014
Numbers k such that 16*k+1 is a fourth power. - Bruno Berselli, May 29 2018
The row sums of "Floyd's Triangle", which is a triangular array of natural numbers beginning with the number 1, produce the sequence A006003. A006003 can be bisected to get the Rhombic Dodecahedron Sequence A005917, whose n-th partial sum is n^4, and A317297, whose n-th partial sum is a(n). Interleave n^4 or A000583 back with {a(n)} to get A011863, whose first differences are A019298. Finally, A011863(n)-A011863(n-2) = A006003(n-1). - Bruce J. Nicholson, Dec 22 2019

			0^(1/4) = 0.000...; 1^(1/4) = 1.000...
5^(1/4) = 1.495...; 6^(1/4) = 1.565...
39^(1/4) = 2.499...; 40^(1/4) = 2.514...

Clark Kimberling, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A123865, A219085, A234459, A000217, A001844.

Cf. A317297, A006003, A005917, A000583, A011863, A019298.

A219086:=n->floor((n + (1/2))^4); seq(A219086(n), n=0..50); # Wesley Ivan Hurt, Apr 05 2014

Table[Floor[(n + 1/2)^4], {n, 0, 100}]
LinearRecurrence[{5,-10,10,-5,1},{0,5,39,150,410},40] (* Harvey P. Dale, Jan 15 2023 *)

a(n)=floor((n + 1/2)^4) \\ Charles R Greathouse IV, Apr 15 2014

G.f.: (5*x^3 + 14*x^2 + 5*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (2*n^4 + 4*n^3 + 3*n^2 + n)/2. - J. M. Bergot, Apr 05 2014
a(n) = Sum_{i=0..n} i*(4*i^2 + 1) = n*(n + 1)*(2*n^2 + 2*n + 1)/2. - Bruno Berselli, Feb 09 2017
a(n) = lcm((2*n + 1)^2 - 1, (2*n + 1)^2 + 1)/8 for n>=1. - Lechoslaw Ratajczak, Mar 26 2017
a(n) = A000217(n) * A001844(n). - Bruce J. Nicholson, May 14 2017
E.g.f.: (1/2)*exp(x)*x*(10 + 29*x + 16*x^2 + 2*x^3). - Stefano Spezia, Dec 27 2019
a(n) = ((2*n+1)^4 - 1)/16. - Jianing Song, Jan 03 2023
Sum_{n>=1} 1/a(n) = 6 - 2*Pi*tanh(Pi/2). - Amiram Eldar, Jan 08 2023

A234460 Imaginary part of the product of all the integer complex numbers in the square [1,1] to [n,n].

Original entry on oeis.org

1, 0, 7800, 0, 2787453552000000, 0, 3108366378804858902744832000000000000, 0, 165290679439545659068950724771043004678057040281600000000000000000000, 0

Offset: 1

Jon Perry, Dec 26 2013

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..28

Cf. A000142, A234459.

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 < 10; i++) {
c = cNumber(1, 0);
for (j = 1; j <= i; j++)
for (k = 1; k <= i; k++)
c = cMult(c, cNumber(j, k));
document.write(c + "
");
}

Table[Im[Times@@Flatten[Table[a + b I, {a, n}, {b, n}]]], {n, 20}]

a(n) = imag(prod(i=1, n, prod(j=1, n, i+I*j))); \\ Michel Marcus, Dec 31 2013

For n even, a(n) = 0, and for n odd, a(n) = A234459(n). - Michel Marcus, Dec 31 2013

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

cp's OEIS Frontend

A219086 a(n) = floor((n + 1/2)^4).

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A234460 Imaginary part of the product of all the integer complex numbers in the square [1,1] to [n,n].

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

JavaScript

Mathematica

PARI

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