A047220 -id:A047220 - cp's OEIS frontend

A319950 a(n) = Product_{i=1..n} floor(5*i/3).

1, 3, 15, 90, 720, 7200, 79200, 1029600, 15444000, 247104000, 4447872000, 88957440000, 1868106240000, 42966443520000, 1074161088000000, 27928188288000000, 781989272064000000, 23459678161920000000, 727250023019520000000, 23999250759644160000000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

If p > 3 and gcd(p,3)=1 then Product_{i=1..n} floor(i*p/3) ~ (p/3)^n * n! * 2*Pi * 3^(1/p - 1/2) / (c(p) * n^(1/p)), where
c(p) = Gamma(2/3 - 2/(3*p)) * Gamma(1/3 - 1/(3*p)) if mod(p, 3) = 1,
c(p) = Gamma(1/3 - 2/(3*p)) * Gamma(2/3 - 1/(3*p)) if mod(p, 3) = 2.
In general, if q > 1, p > q and gcd(p,q)=1, then Product_{i=1..n} floor(i*p/q) ~ c(p,q) * (p/q)^n * n! / n^((q-1)/(2*p)), where c(p,q) is a constant.

Cf. A010786, A047220, A180736, A275062, A319948, A319949, A317980.

Table[Product[Floor[i*5/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(15*n - 32)*a[n] == 675*(n-2)*a[n-1] + 15*(75*n^2 - 255*n + 194)*a[n-2] + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a[n-3], a[1]==1, a[2]==3, a[3]==15}, a, {n, 1, 20}]

a(n) = prod(i=1, n, (5*i)\3); \\ Michel Marcus, Oct 03 2018

a(n) ~ (5/3)^n * n! * 2*Pi / (3^(3/10) * Gamma(1/5) * Gamma(3/5) * n^(1/5)).

Recurrence: 27*(15*n - 32)*a(n) = 675*(n-2)*a(n-1) + 15*(75*n^2 - 255*n + 194)*a(n-2) + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a(n-3).

A222170 a(n) = n^2 + 2*floor(n^2/3).

Original entry on oeis.org

0, 1, 6, 15, 26, 41, 60, 81, 106, 135, 166, 201, 240, 281, 326, 375, 426, 481, 540, 601, 666, 735, 806, 881, 960, 1041, 1126, 1215, 1306, 1401, 1500, 1601, 1706, 1815, 1926, 2041, 2160, 2281, 2406, 2535, 2666, 2801, 2940, 3081, 3226, 3375, 3526, 3681, 3840

Offset: 0

Bruno Berselli, Aug 08 2013

Also, a(n) = n^2 + floor(2*n^2/3), since 2*floor(n^2/3) = floor(2*n^2/3).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tadeusz Dorozinskis, Pentagonpolyhedra Doro
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Subsequence of A008851.

Cf. A004773 (numbers of the type n+floor(n/3)), A008810 (numbers of the type n^2-2*floor(n^2/3)), A047220 (numbers of the type n+floor(2*n/3)), A184637 (numbers of the type n^2+floor(n^2/3), except the first two).

Magma
```
[n^2+2*Floor(n^2/3): n in [0..50]];
```

I:=[0,1,6,15,26]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Table[n^2 + 2 Floor[n^2/3], {n, 0, 50}]
CoefficientList[Series[x (1 + x) (1 + 3 x + x^2) / ((1 + x + x^2) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 6, 15, 26}, 50] (* Hugo Pfoertner, Jan 17 2023 *)

G.f.: x*(1+x)*(1 + 3*x + x^2)/((1 + x + x^2)*(1-x)^3).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = floor(5*n^2/3). - Wesley Ivan Hurt, Mar 16 2015
a(n) = a(n-3) + 5*(2n-3) [Tadeusz Dorozinski]. - Eduard Baumann, Jan 18 2023

A272915 a(n) = n + floor(5*n/6).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 66, 67, 69, 71, 73, 75, 77, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 111, 113, 115, 117, 119

Offset: 0

Bruno Berselli, Jun 15 2016

Equivalently, numbers congruent to {0, 1, 3, 5, 7, 9} mod 11.

In general, n + floor((k-1)*n/k) provides the numbers congruent to {0, 1, 3, 5, ..., 2*k-3} mod (2*k-1) for k>1.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. similar sequences with formula n+floor((k-1)*n/k): A032766 (k=2), A047220 (k=3), A047392 (k=4), A187318 (k=5).

Magma
```
[n+Floor(5*n/6): n in [0..70]];
```
Mathematica
```
Table[n + Floor[5 n/6], {n, 0, 70}]
```
Maxima
```
makelist(n+floor(5*n/6), n, 0, 70);
```
PARI
```
vector(70, n, n--; n+floor(5*n/6))
```
Python
```
[n+int(5*n/6) for n in range(70)]
```
Sage
```
[n+floor(5*n/6) for n in range(70)];
```

G.f.: x*(1 + 2 x + 2 x^2 + 2 x^3 + 2 x^4 + 2 x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5)).
a(n) = a(n-1) + a(n-6) - a(n-7).
a(6*k + r) = 11*k + 2*r - (1 - (-1)^a(r))/2, with r = 0..5.

cp's OEIS Frontend

A319950 a(n) = Product_{i=1..n} floor(5*i/3).

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A222170 a(n) = n^2 + 2*floor(n^2/3).

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Magma

Magma

Mathematica

Formula

A272915 a(n) = n + floor(5*n/6).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Python

Sage

Formula