A002265 -id:A002265 - cp's OEIS frontend

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

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

A057362 a(n) = floor(5*n/13).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 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 (1,0,0,0,0,0,0,0,0,0,0,0,1,-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(5*n/13): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Table[Floor[5*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},{0,0,0,1,1,1,2,2,3,3,3,4,4,5},80] (* Harvey P. Dale, Dec 12 2021 *)

a(n)=5*n\13 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: x^3*(1 + x^3 + x^5 + x^8 + x^10) / ( (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]

Sum_{n>=3} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/5 + arccosh(7/2)/(2*sqrt(5)) + log(2)/5. - Amiram Eldar, Sep 30 2022

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

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, 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]

A131295 a(n)=ds_4(a(n-1))+ds_4(a(n-2)), a(0)=0, a(1)=1; where ds_4=digital sum base 4.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3

Offset: 0

Hieronymus Fischer, Jun 27 2007

The digital sum analog (in base 4) of the Fibonacci recurrence.
When starting from index n=3, periodic with Pisano period A001175(3)=8.
Also a(n)==A004090(n) modulo 3 (A004090(n)=digital sum of Fib(n)).
For general bases p>2, the inequality 2<=a(n)<=2p-3 holds for n>2. Actually, a(n)<=5=A131319(4) for the base p=4.
a(n) and Fib(n)=A000045(n) are congruent modulo 3 which implies that (a(n) mod 3) is equal to (Fib(n) mod 3)=A082115(n-1) (for n>0). Thus (a(n) mod 3) is periodic with the Pisano period = A001175(3)=8 too. - Hieronymus Fischer

			a(8)=3, since a(6)=5=11(base 4), ds_4(5)=2,
a(7)=4=10(base 4), ds_4(4)=1 and so a(8)=2+1.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for Colombian or self numbers and related sequences

Cf. A000045, A010073, A010074, A010075, A010076, A010077, A131294, A131296, A131297, A131318, A131319, A131320.

nxt[{a_,b_}]:={b,Total[IntegerDigits[a,4]]+Total[IntegerDigits[b,4]]}; NestList[ nxt,{0,1},110][[All,1]] (* Harvey P. Dale, Jul 30 2018 *)

a(n)=a(n-1)+a(n-2)-3*(floor(a(n-1)/4)+floor(a(n-2)/4)).
a(n)=floor(a(n-1)/4)+floor(a(n-2)/4)+(a(n-1)mod 4)+(a(n-2)mod 4).
a(n)=A002265(a(n-1))+A002265(a(n-2))+A010873(a(n-1))+A010873(a(n-2)).
a(n)=Fib(n)-3*sum{1A000045(n).

A132270 a(n) = floor((n^7-1)/(7*n^6)), which is the same as integers repeated 7 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10

Offset: 1

Mohammad K. Azarian, Nov 06 2007

Wolfgang Hornfeck, Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals, Acta Cryst. (2023) Vol. 79, Part 6, 570-586.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A004526 ([n/2]), A002264 ([n/3]), A002265 ([n/4]), A002266 ([n/5]), A054895.
Cf. A152467 ([n/6]), A132292 ([(n-1)/8]).
Cf. A002162.

A132270:=n->floor((n-1)/7); seq(A132270(n), n=1..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[(n - 1)/7], {n, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[PadRight[{},7,n],{n,0,10}]//Flatten (* or *) LinearRecurrence[ {1,0,0,0,0,0,1,-1},{0,0,0,0,0,0,0,1},100] (* Harvey P. Dale, Jun 08 2017 *)

a(n)=(n-1)\7 \\ Charles R Greathouse IV, Dec 10 2013

a(n) = floor((n^7-n^6)/(7*n^6-6*n^5)). - Mohammad K. Azarian, Nov 08 2007
G.f.: x^8/(1-x-x^7+x^8). - Robert Israel, Feb 02 2015
a(n) = a(n-1)+a(n-7)-a(n-8). - Wesley Ivan Hurt, May 03 2021
a(n) = floor((n-1)/7). - M. F. Hasler, May 19 2021
Sum_{n>=8} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Sep 30 2022

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A185204 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 2, 1, 1, 4, 1, 1, 0, 4, 0, 1, 1, 6, 2, 1, 0, 7, 0, 1, 1, 10, 9, 1, 1, 0, 11, 0, 0, 1, 1, 15, 35, 2, 1, 0, 17, 0, 2, 1, 1, 23, 177, 15, 1, 1, 0, 26, 0, 35, 0, 1, 1, 33, 1153, 247, 1, 1, 0, 38, 0, 1692, 0, 1, 1, 49, 10341, 17409, 8, 1

Offset: 1

Jason Kimberley, Feb 22 2011

For n >= 0 and 0 <= k <= A002265(n).

			0;
1;
1;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1, 1;
1, 0, 1;
1, 1, 2;
1, 0, 2;
1, 1, 4, 1;
1, 0, 4, 0;
1, 1, 6, 2;
1, 0, 7, 0;
1, 1, 10, 9, 1;
1, 0, 11, 0, 0;
1, 1, 15, 35, 2;
1, 0, 17, 0, 2;

Jason Kimberley, Table of i, a(i)=D(n,k) for i = 1..147 (n = 1..32)
Jason Kimberley, Incomplete table of i, n, k, D(n,k)=a(i) for n = 1..38 (i = 1..209)
Jason Kimberley, Disconnected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), this sequence (triangle); specified degree k: A185224 (k=2), A185234 (k=3), A185244 (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).

The b-file corrected and a-file expanded by the author, Jan 19 2013

A208970 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first and second differences in -k..k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 4, 3, 2, 1, 2, 2, 8, 9, 8, 2, 1, 2, 5, 11, 19, 29, 15, 4, 1, 3, 5, 18, 40, 90, 87, 42, 4, 1, 3, 5, 24, 77, 221, 371, 325, 94, 7, 1, 3, 8, 35, 130, 495, 1185, 1755, 1148, 246, 7, 1, 3, 8, 45, 213, 967, 3186, 6883, 8092, 4168, 613, 14, 1

Offset: 1

R. H. Hardin, Mar 03 2012

Table starts
.1..1...1....1....1.....1.....1......1......1......1.......1.......1......1
.1..1...1....2....2.....2.....2......3......3......3.......3.......4......4
.1..1...2....2....2.....5.....5......5......8......8.......8......13.....13
.1..3...4....8...11....18....24.....35.....45.....61......76......98....119
.1..3...9...19...40....77...130....213....325....484.....687.....956...1294
.2..8..29...90..221...495...967...1801...3093...5050....7921...11994..17488
.2.15..87..371.1185..3186..7425..15658..30368..55222...95087..156612.248194
.4.42.325.1755.6883.21830.58791.140429.304536.612054.1154448.2066531

			Some solutions for n=5, k=5:
.-2...-2...-1...-3...-2...-1...-2...-2...-2...-1...-2...-1...-3...-1...-2...-2
.-2...-1....0...-2...-1...-1...-1....0....0....0....0....0...-1....0....0...-2
..0....1....1....2....2....0....2...-1....2...-1....1....0....2....0....0...-1
..2....2...-1....2....1....2....0....2...-1....0....1....0....3....1....2....3
..2....0....1....1....0....0....1....1....1....2....0....1...-1....0....0....2

R. H. Hardin, Table of n, a(n) for n = 1..184

Row 2 is A002265(n+4).

Row 3 is A000982(floor(n/3)+1).

Empirical for row n:
n=2: a(k) = a(k-1) + a(k-4) - a(k-5).
n=3: a(k) = a(k-1) + a(k-3) - a(k-4) + a(k-6) - a(k-7) - a(k-9) + a(k-10).
n=4: a(k) = 2*a(k-1) + a(k-2) - 4*a(k-3) + a(k-4) + 2*a(k-5) - a(k-6).

cp's OEIS Frontend

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

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A057362 a(n) = floor(5*n/13).

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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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A131295 a(n)=ds_4(a(n-1))+ds_4(a(n-2)), a(0)=0, a(1)=1; where ds_4=digital sum base 4.