A057355 -id:A057355 - cp's OEIS frontend

A057363 a(n) = floor(8*n/13).

0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 41, 42, 43, 43, 44, 44

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Note that 20 appears twice. Different from A005206, A060143.

[Floor(8*n/13): n in [0..50]]' // G. C. Greubel, Nov 02 2017

Table[Floor[8*n/13], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,1,-1},{0,0,1,1,2,3,3,4,4,5,6,6,7,8},80] (* Harvey P. Dale, Jul 21 2020 *)

a(n)=8*n\13 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-13) - a(n-14).

G.f.: x^2*(1+x)*(x^2 - x + 1)*(x^8 + x^7 + x^2 + 1)/( (x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x-1)^2 ). [Numerator corrected Feb 20 2011]

A057364 a(n) = floor(8*n/21).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[floor(8*n/21): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Table[Floor[8 n/21],{n,0,80}] (* Harvey P. Dale, Jun 14 2011 *)

a(n)=8*n\21 \\ Charles R Greathouse IV, Jul 07 2011

a(n) = a(n-1) + a(n-21) - a(n-22).

G.f.: x^3*(1+x)*(x^4 - x^3 + x^2 - x + 1)*(x^13 + x^11 + x^3 + 1) / ( (1 + x + x^2)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1)*(x-1)^2 ). [Numerator corrected by R. J. Mathar, Feb 20 2011]

A057365 a(n) = floor(13*n/21).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(12*n/21): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Table[Floor[13*n/21], {n,0,50}] (* G. C. Greubel, Nov 02 2017 *)

13*n\21 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-21) - a(n-22).

G.f.: x^2*(1 + x^2 + x^3 + x^5 + x^7 + x^8 + x^10 + x^11 + x^13 + x^15 + x^16 + x^18 + x^19)/( (1+x+x^2)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1)*(x-1)^2 ). [Numerator corrected Feb 20 2011]

A057366 a(n) = floor(7*n/19).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, 28

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,1,-1).

Similar pattern in Hebrew leap years A057349. Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(7*n/19): n in [0..50]]; // G. C. Greubel, Nov 03 2017

Table[Floor[7*n/19], {n,0,50}] (* G. C. Greubel, Nov 03 2017 *)

a(n)=7*n\19 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-19) - a(n-20).

G.f.: x^3*(x^2-x+1)*(x^14 + x^13 + x^12 - x^10 + x^8 + x^7 + x^6 + x + 1)/( (x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x-1)^2 ). [Corrected by R. J. Mathar, Feb 20 2011]

A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 14, 18, 23, 29, 35, 42, 49, 57, 66, 75, 85, 95, 106, 118, 130, 143, 156, 170, 185, 200, 216, 232, 249, 267, 285, 304, 323, 343, 364, 385, 407, 429, 452, 476, 500, 525, 550, 576, 603, 630, 658, 686, 715, 745, 775, 806, 837, 869, 902, 935

Offset: 1

Clark Kimberling, Jul 08 2013

See A227347.

			a(1) = floor(3/5) = 0; a(2) = floor(6/5) = 1; a(3) = a(2) + floor(9/5) = 2; a(4) = a(3) + floor(12/5) = 4.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A227347, A130520, A011858, A033437.

Cf. A057355 (first differences).

z = 150; r = 3/5; k = 1; a[n_] := Sum[Floor[r*x^k], {x, 1, n}]; t = Table[a[n], {n, 1, z}]

a(n) = (3*n^2-n)\10; \\ Kevin Ryde, Mar 15 2022

a = lambda n: n*(3*n-1)//10 # Gennady Eremin, Mar 20 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
G.f.: (x*(1 + x^2 + x^3))/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
According to Wolfram Alpha, a(n) = floor(Re(E(n^2|Pi))) where E(x|m) is the incomplete elliptic integral of the second kind. - Kritsada Moomuang, Jan 28 2022
a(n) = a(n-1) + floor(3*n/5), n > 1. - Gennady Eremin, Mar 15 2022
a(n) = floor(n*(3*n-1)/10). - Kevin Ryde, Mar 15 2022

A327440 a(n) = floor(3*n/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23

Offset: 0

Bruno Berselli, Sep 11 2019

The sequence can be obtained from A008585 by deleting the last digit of each term.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008585.

Similar sequences with the formula floor(k*n/10): A059995 (k=1); A002266 (k=2); A057354 (k=4); A004526 (k=5); A057355 (k=6); A188511 (k=7); A090223 (k=8).

[div(3*n, 10) for n in 0:80] |> println

Table[Floor[3 n/10], {n, 0, 80}]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3}, 80]

PARI
```
vector(80, n, n--; floor(3*n/10))
```

O.g.f.: x^4*(1 + x^3 + x^6)/((1 + x)*(1 - x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) = (x^4 + x^7 + x^10)/(1 - x - x^10 + x^11).

a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10.

cp's OEIS Frontend

A057363 a(n) = floor(8*n/13).

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A057364 a(n) = floor(8*n/21).

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A057365 a(n) = floor(13*n/21).

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A057366 a(n) = floor(7*n/19).

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A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

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A327440 a(n) = floor(3*n/10).

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