A002266 -id:A002266 - cp's OEIS frontend

A010884 Period 5: repeat [1,2,3,4,5].

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130483(n)+n+1. - Hieronymus Fischer, Jun 08 2007

4115/33333 = 0.12345123451234512345... - Eric Desbiaux, Nov 03 2008

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

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266.

Cf. A177038 (decimal expansion of (195+sqrt(65029))/314).

PadRight[{},120,Range[5]] (* Harvey P. Dale, Dec 08 2018 *)

a(n)=n%5+1 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = 1 + (n mod 5). - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
G.f.: (5*x^4+4*x^3+3*x^2+2*x+1)/(1-x^5) = (5*x^6-6*x^5+1)/((1-x^5)*(1-x)^2).
a(n) = A010874(n)+1. (End)
a(n) = a(n-5). - Wesley Ivan Hurt, Jan 15 2022

A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

Original entry on oeis.org

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 355, 385, 416, 448, 481, 516, 552, 589, 627, 666, 707, 749, 792, 836, 881, 928, 976, 1025, 1075, 1126, 1179

Offset: 0

Paul Curtz, Feb 11 2020

This sequence occurs twice as a linear spoke in the hexagonal spiral constructed from A002266:
                       17  17  17  17  17  18  18
                     16  11  11  11  11  12  12  18
                   16  11   6   6   7   7   7  12  18
                 16  10   6   3   3   3   3   7  12  18
               16  10   6   3   1   1   1   4   7  12  19
             16  10   6   2   0   0   0   1   4   8  13  19
               15  10   5   2   0   0   1   4   8  13  19
                 15  10   5   2   2   2   4   8  13  19
                   15   9   5   5   5   4   8  13  19
                     15   9   9   9   9   8  13  20
                       15  14  14  14  14  14  20
a(-1-n) = 0, 1, 4, 8, 13, 19, 26, 35, 45, ... also occurs twice in the same spiral.
Difference table:
  0, 1, 3, 6, 11, 17, 24, 32, 41, 52, ... = a(n)
  1, 2, 3, 5,  6,  7,  8,  9, 11, 12, ... = A047256(n+1)
  1, 1, 2, 1,  1,  1,  1,  2,  1,  1, ... = A130782.
There is no linear spoke with three copies in this spiral. Compare with the spiral illustrated in sequence A330707 and constructed from A002265 where the same spokes occur three times: A006578, A001859 and A077043, essentially. Strictly, three times from 1, 1, 1 for A006578, from 2, 2, 2 for A001859 and from 7, 7, 7 for A077043.

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

Cf. A002266, A008602, A047256, A130782, A330707.

Cf. A001859, A006578, A077043.

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 11, 17, 24}, 45] (* Amiram Eldar, Feb 12 2020 *)

concat(0, Vec(x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^50))) \\ Colin Barker, Feb 11 2020, Apr 24 2020

a(8+n) - a(8-n) = 20*n.

G.f.: x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Feb 11 2020

A343609 a(n) = floor(n/9).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, May 19 2021

Also: Nonnegative integers repeated 9 times (with natural offset 0).

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

Cf. A004526 ([n/2]), A002264 ([n/3]), A002265 ([n/4]), A002266 ([n/5]), A152467 ([n/6]), A132270 ([(n-1)/7]), A132292 ([(n-1)/8]), A059995 ([n/10]), A344420 ([n/11]), A342696 ([n/12]).

Repunits A002275 = A343609 o A011557.

A343609 := n -> iquo(n,9); # illustration: map( A343609, [$0..99] );

A343609[n_] := Floor[n/9]
a[n_] := Quotient[n, 9]; Array[a, 100, 0] (* Amiram Eldar, May 19 2021 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,1,-1},{0,0,0,0,0,0,0,0,0,1},100] (* Harvey P. Dale, Mar 01 2025 *)

PARI
```
apply( A343609(n)=n\9, [0..99])
```

a(n) = A002264(A002264(n)).
a(n) = a(n-1) + a(n-9) - a(n-10), n > 9;
G.f.: x^9/(1 - x - x^9 + x^10).

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Oct 31 2021

This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.
The length of row n is floor(n/2) = A004526(n), for n >= 2.
The row sums give A002541(n). See the formula given there by Wesley Ivan Hurt, May 08 2016.
The columns give the k-fold repeated positive integers k, for k >= 1.

			The irregular triangle T(n, k) begins:
n\k   1 2 3 4 5 6 7 8 9 10 ...
------------------------------
2:    1
3:    2
4:    3 1
5:    4 1
6:    5 2 1
7:    6 2 1
8:    7 3 1 1
9:    8 3 2 1
10:   9 4 2 1 1
11:  10 4 2 1 1
12:  11 5 3 2 1 1
13:  12 5 3 2 1 1
14:  13 6 3 2 1 1 1
15:  14 6 4 2 2 1 1
16:  15 7 4 3 2 1 1 1
17:  16 7 4 3 2 1 1 1
18:  17 8 5 3 2 2 1 1 1
19:  18 8 5 3 2 2 1 1 1
20:  19 9 5 4 3 2 1 1 1  1
...

Karl-Heinz Hofmann, Table of n, a(n) for n = 2..10101

Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

Cf. A002541, A004526, A010766, A348389.

T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 02 2021 *)

def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]
for n in range(2, 21): print(A348388row(n))  # Peter Luschny, Nov 05 2021

T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

A010886 Period 7: repeat [1, 2, 3, 4, 5, 6, 7].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4

Offset: 0

N. J. A. Sloane

Partial sums are given by A130485(n)+n+1. - Hieronymus Fischer, Jun 08 2007

Decimal expansion of 1234567/9999999 = 0.123456712345671234567... - Eric Desbiaux, Nov 03 2008

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

Cf. A002264, A002265, A002266, A004526, A010872, A010873, A010874, A010875, A010876, A010877, A130485.

Cf. A177160 (decimal expansion of (4502+sqrt(29964677))/6961).

&cat [[1, 2, 3, 4, 5, 6, 7]^^20]; // Wesley Ivan Hurt, Jul 18 2016

seq(op([1, 2, 3, 4, 5, 6, 7]), n=0..20); # Wesley Ivan Hurt, Jul 18 2016

PadRight[{}, 100, {1, 2, 3, 4, 5, 6, 7}] (* Wesley Ivan Hurt, Jul 18 2016 *)

a(n)=n%7+1 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = 1 + (n mod 7). - Paolo P. Lava, Nov 21 2006
a(n) = A010876(n) + 1. G.f.: (Sum_{k=0..6} (k+1)*x^k)/(1-x^7). Also (7*x^8-8*x^7+1)/((1-x^7)*(1-x)^2). - Hieronymus Fischer, Jun 08 2007
From Wesley Ivan Hurt, Jul 18 2016: (Start)
a(n) = a(n-7) for n>6.
a(n) = 1 - 6*floor(n/7) + Sum_{k=1..6} floor((n + k)/7). (End)

A131766 a(n) = A131668(n) - (2*n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 18, 36, 54, 72, 180, 378, 576, 774, 972, 2970, 4968, 6966, 8964, 19962, 39960, 59958, 79956, 99954, 299952, 499950, 699948, 899946, 1999944, 3999942, 5999940, 7999938, 9999936, 29999934, 49999932, 69999930, 89999928, 199999926, 399999924, 599999922

Offset: 0

Paul Curtz, Oct 04 2007

Digital sums: 9*A002266.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,100,-200,100).

Cf. A002266, A131668, A131871.

a(30) corrected and more terms from Georg Fischer, Jul 30 2025

A131871 a(n) = A131766(n) / 18.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 10, 21, 32, 43, 54, 165, 276, 387, 498, 1109, 2220, 3331, 4442, 5553, 16664, 27775, 38886, 49997, 111108, 222219, 333330, 444441, 555552, 1666663, 2777774, 3888885, 4999996, 11111107, 22222218, 33333329, 44444440, 55555551, 166666662

Offset: 0

Paul Curtz, Oct 05 2007

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,100,-200,100).

Cf. A002266, A131766.

a(32) corrected and more terms from Georg Fischer, Jul 31 2025

A137935 a(n) = 5n + 26*floor(n/5).

Original entry on oeis.org

0, 5, 10, 15, 20, 51, 56, 61, 66, 71, 102, 107, 112, 117, 122, 153, 158, 163, 168, 173, 204, 209, 214, 219, 224, 255, 260, 265, 270, 275, 306, 311, 316, 321, 326, 357, 362, 367, 372, 377, 408, 413, 418, 423, 428, 459, 464, 469, 474, 479, 510, 515, 520, 525, 530, 561, 566

Offset: 0

William A. Tedeschi, Mar 06 2008

			a(0) = 5(0) + 26*floor(0/5) = 0
a(3) = 5(3) + 26*floor(3/5) = 15

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A002266.

seq(5*n + 26*floor(n/5), n=0..200); # Robert Israel, Apr 02 2017

Python
```
a = lambda n: 5*n + 26*floor(n/5)
```

a(n) = 5n + 26*floor(n/5) = 5n + 26*A002266(n)

G.f.: (5*x+5*x^2+5*x^3+5*x^4+31*x^5)/(1-x-x^5+x^6). - Robert Israel, Apr 02 2017

A175884 Numbers that are congruent to {0, 2, 4, 7, 9} mod 12.

Original entry on oeis.org

0, 2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 40, 43, 45, 48, 50, 52, 55, 57, 60, 62, 64, 67, 69, 72, 74, 76, 79, 81, 84, 86, 88, 91, 93, 96, 98, 100, 103, 105, 108, 110, 112, 115, 117, 120, 122, 124, 127, 129, 132, 134, 136, 139, 141, 144, 146, 148, 151

Offset: 1

Bill Shillito (DMAshura(AT)gmail.com), Oct 08 2010

Key-numbers of the pitches of a major pentatonic scale on a standard chromatic keyboard, with root = 0.

The pentatonic scale can also be obtained by omitting the 4th and 7th notes from the diatonic scale, so a(n) = A083026(A032796(n)). - Federico Provvedi, Sep 10 2022

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Wikipedia, Major pentatonic scale
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A005843, A057354, A002266.

Subset of A083026 with exact index A032796.

Filtered([0..151],n->n mod 12 = 0 or n mod 12 = 2 or n mod 12 = 4 or n mod 12 = 7 or n mod 12 = 9); # Muniru A Asiru, Oct 24 2018

[Floor(12*(n-1)/5): n in [1..100]]; // G. C. Greubel, Oct 23 2018

seq(floor(12*(n-1)/5),n=1..65); # Muniru A Asiru, Oct 24 2018

fQ[n_] := MemberQ[{0, 2, 4, 7, 9}, Mod[n, 12]]; Select[ Range[0, 152], fQ] (* Robert G. Wilson v, Oct 09 2010 *)
Table[2n-1+Floor[(n-4)/5]+Floor[(n-1)/5],{n, 100}] (* Federico Provvedi, Jan 13 2018 *)
Quotient[12(Range[100]-1), 5] (* Federico Provvedi, Oct 19 2018 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,1,0,0,0,1]^n*[0;2;4;7;9;12])[1,1] \\ for offset 0; Charles R Greathouse IV, Jul 06 2017

vector(100, n, floor(12*(n-1)/5)) \\ G. C. Greubel, Oct 23 2018

G.f.: x^2*(2 + 2*x + 3*x^2 + 2*x^3 + 3*x^4) / ((x^4 + x^3 + x^2 + x + 1)*(x-1)^2). - R. J. Mathar, Jul 10 2015
a(n) = 2*n - 1 + floor((n-4)/5) + floor((n-1)/5). - Federico Provvedi, Jan 13 2018
a(n) = floor(12*(n-1)/5). - Federico Provvedi, Oct 19 2018
a(n) = A005843(n) + A057354(n). - Federico Provvedi, Sep 10 2022

Offset change by G. C. Greubel, Oct 23 2018

A008381 floor(n/5)floor((n+1)/5)floor((n+2)/5)*floor((n+3)/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 16, 24, 36, 54, 81, 81, 108, 144, 192, 256, 256, 320, 400, 500, 625, 625, 750, 900, 1080, 1296, 1296, 1512, 1764, 2058, 2401, 2401, 2744, 3136, 3584, 4096, 4096, 4608, 5184, 5832

Offset: 0

N. J. A. Sloane

nonn

a(n) = A002266(n)*A008496(n+1). a(n)= +a(n-1) +4*a(n-5) -4*a(n-6) -6*a(n-10) +6*a(n-11) +4*a(n-15) -4*a(n-16) -a(n-20) +a(n-21). G.f.: x^5*(x^10-2*x^9+4*x^8-4*x^7+8*x^6-8*x^5+8*x^4-4*x^3+4*x^2-2*x+1) * (1+x)^2 / ((x^4+x^3+x^2+x+1)^4 * (1-x)^5). [From R. J. Mathar, Apr 16 2010]

cp's OEIS Frontend

A010884 Period 5: repeat [1,2,3,4,5].

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A332410 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

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A343609 a(n) = floor(n/9).

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A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

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A010886 Period 7: repeat [1, 2, 3, 4, 5, 6, 7].

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Magma

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A131766 a(n) = A131668(n) - (2*n+1).

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A131871 a(n) = A131766(n) / 18.

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A137935 a(n) = 5n + 26*floor(n/5).

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A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

A008381 floor(n/5)floor((n+1)/5)floor((n+2)/5)*floor((n+3)/5).