A004777 -id:A004777 - cp's OEIS frontend

A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90

Offset: 0

N. J. A. Sloane

The sequence b(n) = floor((4/3)*(n+2)) appears as an upper bound in Fijavz and Wood.
Binary expansion does not end in 11.
From Guenther Schrack, May 04 2023: (Start)
The sequence is the interleaving of the sequences A008586, A016813, A016825, in that order.
Let S(n) = a(n) + a(n+1) + a(n+2). Then floor(S(n)/3) = A042968(n+1), round(S(n)/3) = a(n+1), ceiling(S(n)/3) = A042965(n+2). (End)

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Gasper Fijavz and David R. Wood, Graph Minors and Minimum Degree, arXiv:0812.1064 [math.CO], 2008.
Niall Graham and Frank Harary, Edge Sums of Hypercubes, Bull. Irish Math. Soc., Vol. 21 (1988), pp. 8-12.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A177702 (first differences), A000969 (partial sums).
Cf. A032766, this sequence, A001068, A047226, A047368, A004777.
Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.
Cf. A002264, A004396, A004523, A004767, A004772, A008586, A010872, A016813, A016825, A042965, A042968.

[n: n in [0..100] | n mod 4 in [0..2]]; // Vincenzo Librandi, Dec 23 2010

seq(floor(n/3)+n,n=0..68); # Gary Detlefs, Mar 20 2010

f[n_] := Floor[4 n/3]; Array[f, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
fQ[n_] := Mod[n, 4] != 3; Select[ Range[0, 90], fQ] (* Robert G. Wilson v, Dec 24 2010 *)
a[0] = 0; a[n_] := a[n] = a[n - 1] + 2 - If[ Mod[a[n - 1], 4] < 2, 1, 0]; Array[a, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
CoefficientList[ Series[x (1 + x + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 68}], x] (* Robert G. Wilson v, Dec 24 2010 *)

a(n)=4*n\3 \\ Charles R Greathouse IV, Sep 27 2012

G.f.: x*(1+x+2*x^2)/((1-x)*(1-x^3)).
a(0) = 0, a(n+1) = a(n) + a(n) mod 4 + 0^(a(n) mod 4). - Reinhard Zumkeller, Mar 23 2003
a(n) = A004396(n) + A004523(n); complement of A004767. - Reinhard Zumkeller, Aug 29 2005
a(n) = floor(n/3) + n. - Gary Detlefs, Mar 20 2010
a(n) = (12*n-3+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
E.g.f.: (3*exp(x)*(4*x - 1) + exp(-x/2)*(3*cos((sqrt(3)*x)/2) + sqrt(3)*sin((sqrt(3)*x)/2)))/9. - Stefano Spezia, Jun 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + sqrt(2)*log(sqrt(2)+2)/4 + (2-sqrt(2))*log(2)/8. - Amiram Eldar, Dec 05 2021
From Guenther Schrack, May 04 2023: (Start)
a(n) = (12*n - 3 +  w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2.
a(n) = 2*floor(n/3) + floor((n+1)/3) + floor((n+2)/3).
a(n) = (4*n - n mod 3)/3.
a(n) = a(n-3) + 4.
a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = 4*A002264(n) + A010872(n).
a(n) = A042968(n+1) - 1.
(End)

A047226 Numbers that are congruent to {0, 1, 2, 3, 4} mod 6; a(n)=floor(6(n-1)/5).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79

Offset: 1

N. J. A. Sloane

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

Cf. A032766, A004773, A001068, this, A047368, A004777, A248375.

[n: n in [0..100] | n mod 6 in [0..4]]; // Vincenzo Librandi, Jan 06 2013

A047226 := proc(n)
    option remember;
    if n <= 6 then
        op(n,[0,1,2,3,4,6]) ;
    else
        procname(n-1)+procname(n-5)-procname(n-6) ;
    end if;
end proc: # R. J. Mathar, Jul 25 2013

Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

G.f.: x^2*(1+x+x^2+x^3+2*x^4) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Sep 17 2015, Jul 16 2013: (Start)
a(n) = floor( 6*(n-1)/5 ).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>6.
a(n) = n - 1 + floor((n-1)/5). (End)
Sum_{n>=2} (-1)^n/a(n) = (9-4*sqrt(3))*Pi/36 + log(2+sqrt(3))/(2*sqrt(3)) + log(2)/6. - Amiram Eldar, Dec 17 2021

Explicit formula added to definition by M. F. Hasler, Oct 05 2014

A047368 Numbers that are congruent to {0, 1, 2, 3, 4, 5} mod 7; a(n)=floor(7(n-1)/6).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77

Offset: 1

N. J. A. Sloane

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

Cf. A001068, A004773, A004777, A032766, A047226, A248375.

[n : n in [0..100] | n mod 7 in [0..5]]; // Wesley Ivan Hurt, Jun 15 2016

A047368:=n->(42*n-57-3*cos(Pi*n)-4*sqrt(3)*cos((4*n+1)*Pi/6)-12*sin((1-2*n)*Pi/6))/36: seq(A047368(n), n=1..100); # Wesley Ivan Hurt, Jun 15 2016

Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4, 5}, Mod[#, 7]] &] (* Wesley Ivan Hurt, Jun 15 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 5, 7}, 100] (* Vincenzo Librandi, Jun 16 2016 *)

a(n)=(n-1)*7\6 \\ M. F. Hasler, Oct 05 2014

G.f.: x^2*(1+x+x^2+x^3+x^4+2*x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = (42*n-57-3*cos(Pi*n)-4*sqrt(3)*cos((4*n+1)*Pi/6)-12*sin((1-2*n)*Pi/6))/36.
a(6k) = 7k-2, a(6k-1) = 7k-3, a(6k-2) = 7k-4, a(6k-3) = 7k-5, a(6k-4) = 7k-6, a(6k-5) = 7k-7. (End)

Crossrefs and explicit formula in name added by M. F. Hasler, Oct 05 2014

A059561 Beatty sequence for log(Pi).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81

Offset: 1

Mitch Harris, Jan 22 2001

a(n) is the largest integer m such that e^m < Pi^n. - Stanislav Sykora, May 29 2015

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059562.
Cf. A000796 (Pi), A001113 (e), A053510 (log(Pi)).
Cf. A022932 (characteristic function).

Floor[Range[100]*Log[Pi]] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(Pi); for (n = 1, 2000, write("b059561.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = A004777(n+1), 1 <= n < 83. - R. J. Mathar, Oct 05 2008

a(n) = floor(n*log(Pi)). - Michel Marcus, Jan 04 2015

A248375 a(n) = floor(9*n/8).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 0

M. F. Hasler, Oct 05 2014

Also: numbers not congruent to 8 (mod 9), or numbers whose base-9 expansion does not end in the digit "8".

Paz proves that for all n>0 there is a prime in Breusch's interval [n; a(n+3)], cf A248371.

G. A. Paz, On the Interval [n; 2n]: Primes, Composites and Perfect Powers, Gen. Math. Notes 15 no. 1 (2013), 1-15.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A032766, A004773, A001068, A047226, A047368, A004777, A168183, A281899 (subsequence).

[Floor(9*n/8): n in [0..90]]; // Bruno Berselli, Oct 06 2014

Table[Floor[9 n/8], {n, 0, 90}] (* Bruno Berselli, Oct 06 2014 *)

PARI
```
a(n)=9*n\8
```

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + 2*x^7) / ((1 + x)*(1 - x)^2*(1 + x^2)*(1 + x^4)). [Bruno Berselli, Oct 06 2014]
a(n) = n + floor(n/8) = a(n-1) + a(n-8) - a(n-9). [Bruno Berselli, Oct 06 2014]
a(n) = A168183(n+1) - 1. - Philippe Deléham, Dec 05 2013

A250482 Numbers of the form 2^x + y^2, with x and y >=0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 20, 24, 25, 26, 27, 29, 32, 33, 36, 37, 38, 40, 41, 44, 48, 50, 51, 52, 53, 57, 64, 65, 66, 68, 72, 73, 80, 81, 82, 83, 85, 89, 96, 97, 100, 101, 102, 104, 108, 113, 116, 122, 123, 125, 128, 129, 132, 137, 144

Offset: 1

Vincenzo Librandi, Nov 24 2014

nonn

No terms are congruent to 7 mod 8: subsequence of A004777.

			12 is in this sequence because 2^3+2^2 = 12.
51 is in this sequence because 2^1+7^2 = 51.

Vincenzo Librandi, Table of n, a(n) for n = 1..2500

Cf. A004777, A250481.

Cf. sequences of the type k^x+y^k: this sequence (k=2), A250483 (k=3), A250545 (k=4), A250546 (k=5), A250547 (k=6), A250715 (k=7).

nn=15; Union[Select[Flatten[Table[2^x + y^2, {x, 0, nn}, {y, 0, nn}]], # <=nn^2 &]]

isok(n) = {k=0; while (2^k <= n, if (issquare(n - 2^k), return(1)); k++;); return (0);} \\ Michel Marcus, Nov 24 2014

A047421 Floor(8n/7).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75

Offset: 0

N. J. A. Sloane

Up to the offset identical to A004777, cf formula. - M. F. Hasler, Oct 06 2014

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

Cf. A032766, A004773, A001068, A047226, A047368, A004777.

Table[Floor[8 n/7], {n, 0, 80}] (* Bruno Berselli, Oct 06 2014 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,8},70] (* Harvey P. Dale, Mar 06 2016 *)

PARI
```
a(n)=n\7+n \\ M. F. Hasler, Oct 06 2014
```

a(n) = A004777(n+1). - M. F. Hasler, Oct 06 2014
G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6) / (1 - x - x^7 + x^8). [Bruno Berselli, Oct 06 2014]
a(n) = n + floor(n/7) = a(n-1) + a(n-7) - a(n-8). [Bruno Berselli, Oct 06 2014]

More terms from Ray Chandler, Sep 05 2004

Restored to version of early 2008 by M. F. Hasler, Oct 06 2014

cp's OEIS Frontend

A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

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A047226 Numbers that are congruent to {0, 1, 2, 3, 4} mod 6; a(n)=floor(6(n-1)/5).

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A047368 Numbers that are congruent to {0, 1, 2, 3, 4, 5} mod 7; a(n)=floor(7(n-1)/6).

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A059561 Beatty sequence for log(Pi).

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A248375 a(n) = floor(9*n/8).

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A250482 Numbers of the form 2^x + y^2, with x and y >=0.

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A047421 Floor(8n/7).

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