A057359 -id:A057359 - 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]

A355336 Number of unlabeled n-node graphs with the largest possible bipartite dimension (or biclique covering number).

Original entry on oeis.org

1, 1, 1, 6, 7, 5, 1, 372, 326

Offset: 1

Pontus von Brömssen, Jun 29 2022

Let b(n) be the largest bipartite dimension of an n-node graph. Then b(n) >= A057359(n). If, for all n >= 8, there exists a disconnected n-node graph with bipartite dimension b(n), then b(n) = A057359(n) for all n >= 1. Proof: Since the bipartite dimension of a graph equals the sum of the bipartite dimensions of its connected components, we have that b(n) >= max_{k=1..n-1} b(k)+b(n-k) (i.e., the sequence is superadditive), with equality if there exists a disconnected n-node graph with bipartite dimension b(n). It is easily checked that A057359(n) = max_{k=1..n-1} A057359(k)+A057359(n-k) for n >= 8. Since b(n) = A057359(n) for 1 <= n <= 7 (checked by brute force), the result follows.
Since (b(n)) is superadditive, it follows from Fekete's subadditive lemma that the limit of b(n)/n exists and equals the supremum of b(n)/n. It is easy to see that b(n) <= b(n-1) + 1, so this limit is between 5/7 = b(7)/7 and 1.
The numbers of disconnected n-node graphs with bipartite dimension b(n) for 1 <= n <= 9 are 0, 0, 0, 2, 1, 1, 0, 12, 6.

Wikipedia, Bipartite dimension

Cf. A057359, A355334, A355335.

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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A355336 Number of unlabeled n-node graphs with the largest possible bipartite dimension (or biclique covering number).

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