A047217 -id:A047217 - cp's OEIS frontend

A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

1, 3, 2, 6, 5, 4, 11, 10, 8, 7, 20, 18, 15, 13, 9, 35, 31, 26, 23, 16, 12, 60, 53, 45, 40, 28, 21, 14, 101, 90, 76, 68, 48, 36, 25, 17, 170, 151, 128, 115, 81, 61, 43, 30, 19, 285, 253, 215, 193, 136, 103, 73, 51, 33, 22, 476, 423, 360, 323, 228, 173, 123

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....3....6....11...20
2....5....10...18...31
4....8....15...26...45
7....13...23...40...68
9....16...28...48...81
12...21...36...61...103

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047211, A047220, A191722, A191730, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047220 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
  (* A191740 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191740  *)

A047464 Numbers that are congruent to {0, 2, 4} mod 8.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 18, 20, 24, 26, 28, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 58, 60, 64, 66, 68, 72, 74, 76, 80, 82, 84, 88, 90, 92, 96, 98, 100, 104, 106, 108, 112, 114, 116, 120, 122, 124, 128, 130, 132, 136, 138, 140, 144, 146, 148, 152, 154, 156

Offset: 1

N. J. A. Sloane

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

Cf. A004773, A047395, A047407, A047410.

[n : n in [0..150] | n mod 8 in [0, 2, 4]]; // Wesley Ivan Hurt, Jun 10 2016

A047464:=n->2*(12*n-15-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047464(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Flatten[#+{0,2,4}&/@(8Range[0,20])] (* or *) LinearRecurrence[{1,0,1,-1}, {0,2,4,8}, 80] (* Harvey P. Dale, May 04 2013 *)

a(n) = 2*floor((n-1)/3)+2*n-2. - Gary Detlefs, Mar 18 2010
a(n) = 2*A004773(n-1). G.f.: 2*x^2*(1+x+2*x^2)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Mar 29 2010
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-15-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-4, a(3k-1) = 8k-6, a(3k-2) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + (2-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 19 2021
a(n) = A047217(n)+n-1. - R. J. Mathar, Aug 25 2025

A281899 a(n) = n + 6*floor(n/3).

Original entry on oeis.org

0, 1, 2, 9, 10, 11, 18, 19, 20, 27, 28, 29, 36, 37, 38, 45, 46, 47, 54, 55, 56, 63, 64, 65, 72, 73, 74, 81, 82, 83, 90, 91, 92, 99, 100, 101, 108, 109, 110, 117, 118, 119, 126, 127, 128, 135, 136, 137, 144, 145, 146, 153, 154, 155, 162, 163, 164, 171, 172, 173, 180, 181, 182, 189

Offset: 0

Bruno Berselli, Feb 06 2017

Equivalently, numbers that are congruent to {0, 1, 2} mod 9.
Also numbers m such that floor(m/3) = 3*floor(m/9).
The n-th term is 3*n, 3*n-2 or 3*n-4.
For n > 0, numbers k such that 3 | floor(k/3). - Wesley Ivan Hurt, Dec 01 2020

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

Cf. A002264.
Subsequence of A060464 and A248375.
The first differences are in A105395.
Cf. similar sequences with formula n+i*floor(n/3): A004773 (i=1), A047217 (i=2), A047240 (i=3), A047354 (i=4), A047469 (i=5), this sequence (i=6).
Cf. numbers that are congruent to {0, 1, 2} mod j: the sequences are listed in the previous row for j = 4..9, respectively.

Magma
```
[n+6*(n div 3): n in [0..70]];
```

A281899:=n->n+6*floor(n/3): seq(A281899(n), n=0..100); # Wesley Ivan Hurt, Feb 09 2017

Table[n + 6 Floor[n/3], {n, 0, 70}]
LinearRecurrence[{1,0,1,-1},{0,1,2,9},90] (* Harvey P. Dale, Feb 25 2018 *)

Maxima
```
makelist(n+6*floor(n/3), n, 0, 70);
```

a(n)=n\3*6 + n \\ Charles R Greathouse IV, Feb 07 2017

Python
```
[n+6*int(n/3) for n in range(70)]
```
Sage
```
[n+6*floor(n/3) for n in range(70)]
```

G.f.: x*(1 + x + 7*x^2)/((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = 3*n - 2*(n mod 3). In general, n + 3*h*floor(n/3) = (h+1)*n - h*(n mod 3).
a(n) + a(n+s) = a(2*n+s-1) + 1, where s is nonnegative and not divisible by 3. Example: for s=14, a(n) + a(n+14) = a(2*n+13) + 1; for n=3, a(3) + a(17) = a(19) + 1 = 9 + 47 = 55 + 1 = 56.
a(6*k+r) = 18*k + a(r), where 0 <= r <= 5.
a(n) = 7*A002264(n) + A002264(n+1) + A002264(n+2).

A189786 a(n) = n + [nr/t] + [ns/t]; r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

Original entry on oeis.org

2, 4, 8, 10, 12, 16, 18, 20, 24, 26, 28, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 58, 60, 64, 66, 68, 72, 74, 76, 80, 82, 84, 88, 90, 92, 96, 98, 100, 104, 106, 108, 112, 114, 116, 120, 122, 124, 128, 130, 132, 136, 138, 140, 144, 146, 148, 152, 154, 156, 160, 162, 164, 168, 170, 172, 176, 178, 180, 184, 186, 188, 192, 194, 196, 200, 202, 204

Offset: 1

Clark Kimberling, Apr 27 2011

See A189785.
Conjecture: Sequence consists of all the positive even numbers except numbers of the form 8*x+6, x >= 0. - Harvey P. Dale, Dec 07 2018
Contains numbers like a(143)=382, a(146)=390, a(149)=398, a(152)=406,... which are not in A047464. - R. J. Mathar, Aug 25 2025
For n<143, a(n) = n+A047217(n+1), but then this formula becomes invalid. - R. J. Mathar, Aug 25 2025

Cf. A189785, A189787.

(See A189785.)
With[{t=ArcSin[12/13]},Table[n+Floor[(n*Pi/2)/t]+Floor[(n*ArcSin[5/13])/t],{n,80}]] (* Harvey P. Dale, Dec 07 2018 *)

A272574 a(n) = f(9, f(8, n)), where f(k,m) = floor(m*k/(k-1)).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, May 03 2016

Also, numbers that are congruent to {0..6} mod 9.

The initial terms coincide with those of A037475 and A039111. First disagreement is after 60 (index 48): a(49) = 63, A037475(49) = 81 and A039111(50) = 71.

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

Cf. A248375: f(9,n).

Cf. similar sequences with the formula f(k, f(k-1, n)): A008585 (k=3), A042948 (k=4), A047217 (k=5), A047246 (k=6), A047337 (k=7), A047602 (k=8), this sequence (k=9), A272576 (k=10).

k:=9; f:=func; [f(k,f(k-1,n)): n in [0..70]];

f := (k, m) -> floor(m*k/(k-1)):
a := n -> f(9, f(8, n)):
seq(a(n), n = 0..70); # Peter Luschny, May 03 2016

f[k_, m_] := Floor[m*k/(k-1)];
a[n_] := f[9, f[8, n]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 09 2016 *)
CoefficientList[Series[x (1 + x + x^2 + x^3 + x^4 + x^5 + 3 x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 70}], x] (* or *)
Table[(63 n - 12 - 12 Mod[n, 7] + 2 Mod[-n - 1, 7])/49, {n, 0, 70}] (* Michael De Vlieger, Dec 25 2016 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,9},90] (* Harvey P. Dale, May 08 2018 *)

f = lambda k, m: floor(m*k/(k-1))
a = lambda n: f(9, f(8, n))
[a(n) for n in range(71)] # Peter Luschny, May 03 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + 3*x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8).
a(n) = (63*n - 12 - 12*(n mod 7) + 2*((-n-1) mod 7))/49. - Wesley Ivan Hurt, Dec 25 2016

A298397 Pentagonal numbers divisible by 4.

Original entry on oeis.org

0, 12, 92, 176, 376, 532, 852, 1080, 1520, 1820, 2380, 2752, 3432, 3876, 4676, 5192, 6112, 6700, 7740, 8400, 9560, 10292, 11572, 12376, 13776, 14652, 16172, 17120, 18760, 19780, 21540, 22632, 24512, 25676, 27676, 28912, 31032, 32340, 34580, 35960, 38320, 39772, 42252

Offset: 1

Bruno Berselli, Jan 18 2018

If b(n) is the n-th octagonal number multiple of 32 then a(n) = b(n)/8.

			A000326(8) = 92 is in the sequence because 92 = 4*23.

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

Cf. A000326, A000567.
Subsequence of A047217, A047388.
Cf. pentagonal numbers divisible by k: A014633 (k=2), A268351 (k=3), this sequence (k=4), A117793 (k=5).

List([1..50], n -> 8*n*(3*n-7)-(6*n-7)*(-1)^n+33);

[8*n*(3*n-7)-(6*n-7)*(-1)^n+33: n in [1..50]];

P:=proc(n) local x; x:=n*(3*n-1)/2; if x mod 4=0 then x; fi; end:
seq(P(i),i=0..2*10^2); # Paolo P. Lava, Jan 19 2018

Table[8 n (3 n - 7) - (6 n - 7) (-1)^n + 33, {n, 1, 50}]
(* Second program (using definition): *)
Select[Table[k*(3*k - 1)/2, {k, 0, 200}], Divisible[#, 4]&] (* Jean-François Alcover, Jan 19 2018 *)

makelist(8*n*(3*n-7)-(6*n-7)*(-1)^n+33, n, 1, 50);

vector(50, n, nn; 8*n*(3*n-7)-(6*n-7)*(-1)^n+33)

concat(0, Vec(4*x^2*(3 + 20*x + 15*x^2 + 10*x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jan 20 2018

[8*n*(3*n-7)-(6*n-7)*(-1)^n+33 for n in (1..50)]

O.g.f.: 4*x^2*(3 + 20*x + 15*x^2 + 10*x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (33 - 32*x + 24*x^2)*exp(x) + (7 + 6*x)*exp(-x) - 40.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = 8*n*(3*n - 7) - (6*n - 7)*(-1)^n + 33.
From Colin Barker, Jan 20 2018: (Start)
a(n) = 24*n^2 - 62*n + 40 for n even.
a(n) = 24*n^2 - 50*n + 26 for n odd. (End)

cp's OEIS Frontend

A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A047464 Numbers that are congruent to {0, 2, 4} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A281899 a(n) = n + 6*floor(n/3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Python

Sage

Formula

A189786 a(n) = n + [nr/t] + [ns/t]; r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A272574 a(n) = f(9, f(8, n)), where f(k,m) = floor(m*k/(k-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A298397 Pentagonal numbers divisible by 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

PARI

Sage

Formula