A047226 -id:A047226 - 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)

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

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

A059537 Beatty sequence for zeta(3).

Original entry on oeis.org

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, 80, 81, 82, 84, 85

Offset: 1

Mitch Harris, Jan 22 2001

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 A059538.

Cf. A002117 (zeta(3)).

Floor[Range[100]*Zeta[3]] (* Paolo Xausa, Jul 07 2024 *)

{ default(realprecision, 100); b=zeta(3); for (n = 1, 2000, write("b059537.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 27 2009

a(n) = A047226(n+1), 1<=n<99. - R. J. Mathar, Oct 05 2008

a(n) = floor(n*zeta(3)). - Michel Marcus, Jan 04 2015

A047256 Numbers that are congruent to {0, 1, 2, 3, 5} mod 6.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79

Offset: 1

N. J. A. Sloane

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

Cf. A047226.

Flatten[# + {0, 1, 2, 3, 5} & /@ (6 Range[0, 11])] (* or *)
Select[Range[0, 67], MemberQ[{0, 1, 2, 3, 5}, Mod[#, 6]] &] (* Robert G. Wilson v, Sep 26 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,3,5,6},60] (* Harvey P. Dale, Jul 20 2020 *)

G.f.: x^2*(1+x)*(x^3 + x^2 + 1) / ( (x^4 + x^3 + x^2 + x + 1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
Sum_{n>=2} (-1)^n/a(n) = (3-2*sqrt(3))*Pi/36 + log(2+sqrt(3))/sqrt(3). - Amiram Eldar, Dec 17 2021
a(n) = n + floor(n/5) - 1. - Aaron J Grech, Oct 03 2024

A187343 Rank transform of the sequence floor(6n/5); complement of A187344.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 14, 17, 18, 20, 22, 23, 25, 27, 28, 30, 31, 34, 35, 37, 38, 40, 42, 44, 46, 47, 48, 51, 52, 54, 56, 57, 59, 61, 62, 64, 65, 68, 70, 71, 72, 74, 76, 78, 79, 81, 83, 85, 86, 88, 90, 91, 94, 95, 96, 98, 99, 102, 104, 105, 107, 108, 110, 112, 114, 115

Offset: 1

Clark Kimberling, Mar 08 2011

nonn

See A187224.

Cf. A187224, A187344.

seqA=Table[Floor[6n/5], {n, 1, 220}] (* A047226 *)
seqB=Table[n, {n, 1, 220}];          (* A000027 *)
jointRank[{seqA_, seqB_}]:={Flatten@Position[#1, {, 1}], Flatten@Position[#1, {, 2}]}&[Sort@Flatten[{{#1, 1}&/@seqA, {#1, 2}&/@seqB}, 1]];
limseqU=FixedPoint[jointRank[{seqA, #1[[1]]}]&, jointRank[{seqA, seqB}]][[1]]                  (* A187343 *)
Complement[Range[Length[seqA]], limseqU] (* A187344 *)
(*by Peter J. C. Moses, Mar 07 2011*)

A303602 a(n) = Sum_{k = 0..n} kbinomial(2n+1, k).

Original entry on oeis.org

0, 3, 25, 154, 837, 4246, 20618, 97140, 447661, 2028478, 9070110, 40122028, 175913250, 765561564, 3310623412, 14238676712, 60949133949, 259809601870, 1103420316566, 4670886541308, 19714134528598, 82985455688276, 348481959315660, 1460179866076504, 6106070639175122

Offset: 0

Bruno Berselli, May 09 2018

Second bisection of A185251; the first bisection is A002699.

The terms are not congruent to 5 (mod 6).

Cf. A000346, A002699, A005408, A047226, A164991, A185251.

seq(add(k*binomial(2*n+1,k),k=0..n),n=0..24); # Paolo P. Lava, May 10 2018

Table[Sum[k Binomial[2 n + 1, k], {k, 0, n}], {n, 0, 30}]
CoefficientList[Series[(1 + 4*x - Sqrt[1 - 4*x]) / (2*(1 - 4*x)^2), {x, 0, 25}], x] (* Vaclav Kotesovec, May 10 2018 *)

a(n)=(2*n+1)*(4^n-binomial(2*n,n))/2 \\ Charles R Greathouse IV, Oct 23 2023

[(2*n+1)*(4^n-binomial(2*n,n))/2 for n in (0..30)]

E.g.f.: ((1 + 8*x)*exp(2*x) - (1 + 4*x)*I_0(2*x) - 4*x*I_1(2*x))*exp(2*x)/2, where I_m(.) is the modified Bessel function of the first kind.
From Vaclav Kotesovec, May 10 2018: (Start)
G.f.: (1 + 4*x - sqrt(1 - 4*x)) / (2*(1 - 4*x)^2).
D-finite with recurrence: n*(2*n-1)*a(n) = 2*(2*n+1)*(4*n-3)*a(n-1) - 8*(2*n-1)*(2*n+1)*a(n-2). (End)
a(n) = (2*n + 1)*(4^n - binomial(2*n, n))/2.
a(n+1) - 4*a(n) = A164991(2*n+3).

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

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A059537 Beatty sequence for zeta(3).

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A047256 Numbers that are congruent to {0, 1, 2, 3, 5} mod 6.

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A187343 Rank transform of the sequence floor(6n/5); complement of A187344.

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A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k).

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A047421 Floor(8n/7).

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A303602 a(n) = Sum_{k = 0..n} kbinomial(2n+1, k).