A057357 -id:A057357 - cp's OEIS frontend

A194229 Partial sums of A057357.

0, 1, 2, 4, 6, 9, 12, 15, 19, 23, 28, 33, 39, 45, 51, 58, 65, 73, 81, 90, 99, 108, 118, 128, 139, 150, 162, 174, 186, 199, 212, 226, 240, 255, 270, 285, 301, 317, 334, 351, 369, 387, 405, 424, 443, 463, 483, 504, 525, 546, 568, 590, 613, 636, 660, 684, 708

Offset: 1

Clark Kimberling, Aug 19 2011

			G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + ... - _Michael Somos_, Sep 13 2023

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

Cf. A057357.

r = 3/7;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]    (* A057357 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194229 *)
Table[Sum[Floor[3*k/7], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Nov 03 2017 *)
a[ n_] := Floor[(n^2 + n)*3/14]; (* Michael Somos, Sep 13 2023 *)

concat(0, Vec(x^2*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x+x^2+x^3+x^4 +x^5+x^6)) + O(x^100))) \\ Colin Barker, Jan 09 2016

a(n) = sum(k=1, n, 3*k\7); \\ Michel Marcus, Nov 03 2017

{a(n) = (n^2+n)*3\14}; /* Michael Somos, Sep 13 2023 */

G.f.: x^2*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x+x^2+x^3+x^4+x^5+x^6)). - Colin Barker, Jan 09 2016

G.f.: x^2*(1-x^6) / ((1-x)^2*(1-x^2)*(1-x^7)). - Michael Somos, Sep 13 2023

A057353 a(n) = floor(3n/4).

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53, 54

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.
For n >= 2, a(n) is the number of different integers that can be written as floor(k^2/n) for k = 1, 2, 3, ..., n-1. Generalization of the 1st problem proposed during the 15th Balkan Mathematical Olympiad in 1998 where the question was asked for n = 1998 with a(1998) = 1498. - Bernard Schott, Apr 22 2022
For n > 1, a(n) is also the Hadwiger number of the (n+1)-cycle complement graph (up to at least n = 16). - Eric W. Weisstein, Mar 10 2025

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Balkan Mathematical Olympiad, Problem 1, 15th Balkan Mathematical Olympiad 1998.
Eric Weisstein's World of Mathematics, Cycle Complement Graph.
Eric Weisstein's World of Mathematics, Hadwiger Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences related to Beatty sequences.
Index to sequences related to Olympiads.

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

Cf. A002378, A007310, A057077, A073010, A173562, A182210.

[Floor(3*n/4): n in [0..90]]; // Vincenzo Librandi, Feb 12 2012

Table[Floor[3 n/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Floor[3 Range[0, 20]/4] (* Eric W. Weisstein, Mar 10 2025 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 3, 3}, {0, 20}] (* Eric W. Weisstein, Mar 10 2025 *)
CoefficientList[Series[x^2 (1 + x + x^2)/(1 - x - x^4 + x^5), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 10 2025 *)

a(n)=3*n\4 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3
a(n) = A002378(n) - A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n+1) = A140201(n) - A002265(n+1). - Reinhard Zumkeller, Jan 26 2011
a(n) = n-1 - A002265(n-1) = ( A007310(n) + A057077(n+1) )/4 for n>0. a(n) = a(n-1)+a(n-4)-a(n-5) for n>4. - Bruno Berselli, Jan 28 2011
a(n) = 1/8*(6*n + 2*cos((Pi*n)/2) + cos(Pi*n) - 2*sin((Pi*n)/2) - 3). - Ilya Gutkovskiy, Sep 18 2015
a(4n) = a(4n+1). - Altug Alkan, Sep 26 2015
Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Sep 29 2022

A057354 a(n) = floor(2*n/5).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.
The sequence a(n) can be used in determining confidence intervals for the median of a population. Let Y(i) denote the i-th smallest datum in a random sample of size n from any population of values. When estimating the population median with a symmetric interval [Y(r), Y(n-r+1)], the exact confidence coefficient c for the interval is given by c = Sum_{k=r..n-r} C(n, k)(1/2)^n. If  r = a(n-4), then the confidence coefficient will be (i) at least 0.90 for all n>=7, (ii) at least 0.95 for all n>=35, and (iii) at least 0.99 for all n>=115. To use the sequence, for example, decide on the minimum level of confidence desired, say 95%. Hence use a sample size of 35 or greater, say n=40. We then find a(n-4)=a(36)=14, and thus the 14th smallest and 14th largest values in the sample will form the bounds for the confidence interval. If the exact confidence coefficient c is needed, calculate c = Sum_{k=14..26} C(40,k)(1/2)^40, which is 0.9615226917. - Dennis P. Walsh, Nov 28 2011
a(n+2) is also the domination number of the n-antiprism graph. - Eric W. Weisstein, Apr 09 2016
Equals partial sums of 0 together with 0, 0, 1, 0, 1, ... (repeated). - Bruno Berselli, Dec 06 2016
Euler transform of length 5 sequence [1, 1, 0, -1, 1]. - Michael Somos, Dec 06 2016

			G.f. = x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + ...

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.

Colin Barker, Table of n, a(n) for n = 0..1000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Dennis Walsh, Median estimation with the point-four-n-minus-two rule.
Eric Weisstein's World of Mathematics, Antiprism Graph.
Eric Weisstein's World of Mathematics, Domination Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

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

Cf. A002266.

[2*n div 5: n in [0..80]]; // Bruno Berselli, Dec 06 2016

Table[Floor[2 n/5], {n, 0, 80}] (* Bruno Berselli, Dec 06 2016 *)
a[ n_] := Quotient[2 n, 5]; (* Michael Somos, Dec 06 2016 *)

a(n)=2*n\5 \\ Charles R Greathouse IV, Nov 28 2011

concat(vector(3), Vec(x^3*(1 + x^2) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^80))) \\ Colin Barker, Dec 06 2016.

[int(2*n/5) for n in range(80)] # Bruno Berselli, Dec 06 2016

[floor(2*n/5) for n in range(80)] # Bruno Berselli, Dec 06 2016

G.f.: x^3*(1 + x^2) / ((x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - Numerator corrected by R. J. Mathar, Feb 20 2011
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5. - Colin Barker, Dec 06 2016
a(n) = -a(2-n) for all n in Z. - Michael Somos, Dec 06 2016
a(n) = A002266(2*n). - R. J. Mathar, Jul 21 2020
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2)/2. - Amiram Eldar, Sep 30 2022

A057355 a(n) = floor(3*n/5).

Original entry on oeis.org

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

Offset: 0

Mitch Harris

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

The sequence can be obtained from A008588 by deleting the last digit of each term. - Bruno Berselli, Sep 11 2019

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 (1,0,0,0,1,-1).

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

Cf. A008588.

[3*n div 5: n in [0..80]]; // Bruno Berselli, Dec 07 2016

Table[Floor[3*n/5], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n)=3*n\5 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: x^2*(1 + x^2 + x^3)/((1 - x)*(1 - x^5)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
For all m>=0: a(5m)=0 mod 3; a(5m+1)=0 mod 3; a(5m+2)=1 mod 3; a(5m+3)=1 mod 3; a(5m+4)=2 mod 3.
Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Sep 30 2022

A057356 a(n) = floor(2*n/7).

Original entry on oeis.org

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

Offset: 0

Mitch Harris

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

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

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 (1,0,0,0,0,0,1,-1).

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

Cf. A003881.

[Floor(2*n/7): n in [0..50]]; // G. C. Greubel, Nov 03 2017

Table[Floor[2*n/7], {n,0,50}] (* G. C. Greubel, Nov 03 2017 *)

a(n)=2*n\7 \\ Charles R Greathouse IV, Sep 24 2015

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

Sum_{n>=4} (-1)^n/a(n) = Pi/4 (A003881). - Amiram Eldar, Sep 30 2022

A057359 a(n) = floor(5*n/7).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47, 48, 49, 50, 50, 51

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 (1,0,0,0,0,0,1,-1).

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

Cf. A001622.

[Floor(5*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[5 Range[0,75]/7]  (* Harvey P. Dale, Mar 18 2011 *)

a(n)=5*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f. x^2*(1+x+x^3+x^4+x^5) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). Numerator corrected Feb 20 2011

Sum_{n>=2} (-1)^n/a(n) = sqrt(10-2*sqrt(5))*Pi/10 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 30 2022

A057367 a(n) = floor(11*n/30).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27, 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
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
Index entries for sequences related to Beatty sequences

Similar pattern in Islamic leap years A057347. Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(11*n/30): n in [0..50]]; // G. C. Greubel, Nov 03 2017

A057367:=n->floor(11*n/30); seq(A057367(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[11n/30], {n,0,100}] (* Wesley Ivan Hurt, Oct 29 2013 *)

a(n)=11*n\30 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-30) - a(n-31).

G.f.: x^3*(1 + x^3 + x^6 + x^8 + x^11 + x^14 + x^17 + x^19 + x^22 + x^25 + x^27)/( (1+x)*(1+x+x^2)*(x^2-x+1)*(x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)*(x^8+x^7-x^5-x^4-x^3+x+1)*(x-1)^2 ). [Corrected by R. J. Mathar, Feb 20 2011]

A057358 a(n) = floor(4*n/7).

Original entry on oeis.org

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

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 (1,0,0,0,0,0,1,-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(4*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Table[Floor[4*n/7], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n)=4*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f. x^2*(1+x^2+x^4+x^5) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ) - Numerator corrected by R. J. Mathar, Feb 20 2011

Sum_{n>=2} (-1)^n/a(n) = (Pi - 2*log(sqrt(2)+1))/(4*sqrt(2)). - Amiram Eldar, Sep 30 2022

A057360 a(n) = floor(3*n/8).

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, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 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, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the g.f.) 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.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

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

Cf. A008585, A032766, A132292.

[Floor(3*n/8): n in [0..80]]; // Vincenzo Librandi, Jul 07 2011

A057360:=n->floor(3*n/8): seq(A057360(n), n=0..100); # Wesley Ivan Hurt, May 15 2015

Floor[3 Range[0, 100]/8] (* Wesley Ivan Hurt, May 15 2015 *)
LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{0,0,0,1,1,1,2,2,3},80] (* Harvey P. Dale, Jan 10 2025 *)

a(n)=3*n>>3 \\ Charles R Greathouse IV, Jul 07 2011

G.f.: x^3*(1+x^3+x^5) / ( (1+x)*(x^2+1)*(x^4+1)*(x-1)^2 ).
From Wesley Ivan Hurt, May 15 2015: (Start)
a(n) = a(n-1)+a(n-8)-a(n-9).
a(n) = A132292(A008585(n)), n>0.
a(n) = A002265(A032766(n)). (End)
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Sep 30 2022

Numerator of g.f. corrected by R. J. Mathar, Feb 20 2011

A057361 a(n) = floor(5*n/8).

Original entry on oeis.org

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

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

Cf. A001622.

[Floor(5*n/8): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[(5*Range[0,80])/8] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{0,0,1,1,2,3,3,4,5},80] (* Harvey P. Dale, Jul 18 2013 *)

a(n)=5*n\8 \\ Charles R Greathouse IV, Sep 02 2015

G.f. x^2*(1+x^2+x^3+x^5+x^6) / ( (1+x)*(x^2+1)*(x^4+1)*(x-1)^2 ). - Numerator corrected Feb 20 2011
a(0)=0, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=3, a(6)=3, a(7)=4, a(8)=5, a(n)=a(n-1)+a(n-8)-a(n-9). - Harvey P. Dale, Jul 18 2013
Sum_{n>=2} (-1)^n/a(n) = sqrt(2*(1+1/sqrt(5)))*Pi/10 - log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Sep 30 2022

cp's OEIS Frontend

A194229 Partial sums of A057357.

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A057353 a(n) = floor(3n/4).

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A057354 a(n) = floor(2*n/5).

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A057355 a(n) = floor(3*n/5).

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A057356 a(n) = floor(2*n/7).

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A057359 a(n) = floor(5*n/7).

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