A017185 -id:A017185 - cp's OEIS frontend

A125758 Numbers congruent to 4 or 7 (mod 9).

4, 7, 13, 16, 22, 25, 31, 34, 40, 43, 49, 52, 58, 61, 67, 70, 76, 79, 85, 88, 94, 97, 103, 106, 112, 115, 121, 124, 130, 133, 139, 142, 148, 151, 157, 160, 166, 169, 175, 178, 184, 187, 193, 196, 202, 205, 211, 214, 220, 223, 229, 232, 238, 241, 247, 250, 256, 259, 265, 268

Offset: 1

N. J. A. Sloane and David Applegate, Feb 02 2007

For a given integer m, write its binary representation in reverse order, as in A125626, A125754, etc.; let a 0 mean "halving" and a 1 mean "k -> 3k+1". Then m specifies an operation on real numbers given by k -> f_m(k). Suppose the equation f_m(k) = k has a positive integer solution for some m. Then we conjecture that the values of k are precisely the terms of this sequence.
25 is a term because we have 25 -> 76 -> 38 -> 19 -> 58 -> 29 -> 88 -> 44 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 25.
In other words, we conjecture that this sequence coincides with A125757 sorted and with duplicates removed.

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

Cf. A125626, A125754, A125755, A125756, A125757, A125710, A125711.

Select[Range[300],MemberQ[{4,7},Mod[#,9]]&]  (* Harvey P. Dale, Mar 12 2011 *)

From R. J. Mathar, Apr 03 2009: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 9.
a(n) + a(n+1) = A017185(n).
G.f.: x*(4+3*x+2*x^2)/((1+x)*(x-1)^2). (End)
E.g.f.: 2 + ((9*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 21 2022

A177072 a(n) = (9n+2)(9*n+7).

Original entry on oeis.org

14, 176, 500, 986, 1634, 2444, 3416, 4550, 5846, 7304, 8924, 10706, 12650, 14756, 17024, 19454, 22046, 24800, 27716, 30794, 34034, 37436, 41000, 44726, 48614, 52664, 56876, 61250, 65786, 70484, 75344, 80366, 85550, 90896, 96404, 102074, 107906, 113900, 120056

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 67. - Bruno Berselli, Aug 24 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002061, A017185, A017245, A177059.

I:=[14, 176, 500]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Apr 08 2013

CoefficientList[Series[2(7 + 67 x + 7 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)
Table[(9*n + 2)*(9*n + 7), {n, 0, 40}] (* Amiram Eldar, Feb 19 2023 *)
LinearRecurrence[{3,-3,1},{14,176,500},50] (* Harvey P. Dale, Jun 10 2023 *)

a(n)=(9*n+2)*(9*n+7) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 162*n + a(n-1) with n > 0, a(0)=14.
From Vincenzo Librandi, Apr 08 2013: (Start)
G.f.: 2*(7+67*x+7*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017185(n)*A017245(n).
Sum_{n>=0} 1/a(n) = cot(2*Pi/9)*Pi/45.
Product_{n>=0} (1 - 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(29)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(21)*Pi/18). (End)
E.g.f.: exp(x)*(14 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Edited by N. J. A. Sloane, Jun 22 2010

A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

Original entry on oeis.org

2, 5, 2, 8, 11, 2, 11, 20, 29, 2, 14, 29, 56, 83, 2, 17, 38, 83, 164, 245, 2, 20, 47, 110, 245, 488, 731, 2, 23, 56, 137, 326, 731, 1460, 2189, 2, 26, 65, 164, 407, 974, 2189, 4376, 6563, 2, 29, 74, 191, 488, 1217, 2918, 6563, 13124, 19685, 2, 32, 83, 218, 569, 1460, 3647, 8750, 19685, 39368, 59051, 2

Offset: 0

Wolfdieter Lang, Apr 04 2014

The companion array and triangle for the start numbers M(n, k) is given in A240222.

For the Collatz operations u (for 'up') and d (for 'down') see the comment on A240222, also for links, especially for the M. Trümper paper.

			The rectangular array N(n, k) begins
  n\k 0      1       2       3       4       5 ...
  1:  2      5       8      11      14      17
  2:  2     11      20      29      38      47
  3:  2     29      56      83     110     137
  4:  2     83     164     245     326     407
  5:  2    245     488     731     974    1217
  6:  2    731    1460    2189    2918    3647
  7:  2   2189    4376    6563    8750   10937
  8:  2   6563   13124   19685   26246   32807
  9:  2  19685   39368   59051   78734   98417
  10: 2  59051  118100  177149  236198  295247
  ...
For more columns see the link.
The triangle TN(m, n) begins (zeros are not shown):
  m\n  1  2   3   4    5    6    7 ...
  0:   2
  1:   5  2
  2:   8 11   2
  3:  11 20  29   2
  4:  14 29  56  83    2
  5:  17 38  83 164  245    2
  6:  20 47 110 245  488  731    2
  ...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1, N(1, 0) = TN(0, 1) = 2 with the Collatz sequence [1, 4, 2] of length 3.
n=1, ud, k=2: M(1, 2) = 5, N(1, 2) = TN(2, 1) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1, Ne(2, 0) = TN(1, 2) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.

Wolfdieter Lang, Rectangular array and triangle.
Wolfdieter Lang, On Collatz Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.

Cf. A238475, A238476, A239126, A239127, A240222, A016789 (first row of N), A017185 (second row of N).

The array: N(n, k) = 2 + 3^n*k for n >= 1 and k >= 0.

The triangle: TN(m, n) = N(n,m-n+1) = 2 + 3^n*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.

A373730 Reduced Collatz function R applied to the numbers 6n+1: a(n) = R(6n+1), where R(k) = (3k+1)/2^r, with r as large as possible.

Original entry on oeis.org

1, 11, 5, 29, 19, 47, 7, 65, 37, 83, 23, 101, 55, 119, 1, 137, 73, 155, 41, 173, 91, 191, 25, 209, 109, 227, 59, 245, 127, 263, 17, 281, 145, 299, 77, 317, 163, 335, 43, 353, 181, 371, 95, 389, 199, 407, 13, 425, 217, 443, 113, 461, 235, 479, 61, 497, 253, 515

Offset: 0

Jonas Kaiser, Jun 17 2024

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000265, A017185, A075677, A260658.

A373730[n_] := #/2^IntegerExponent[#, 2] & [9*n + 2];
Array[A373730, 100, 0] (* Paolo Xausa, Aug 19 2024 *)

PARI
```
a(n) = n=9*n+2; n>>valuation(n,2);
```

a(n) = A000265(A017185(n)).

A048502 a(n) = 2^(n-1)(9n-16)+9.

Original entry on oeis.org

1, 2, 13, 53, 169, 473, 1225, 3017, 7177, 16649, 37897, 85001, 188425, 413705, 901129, 1949705, 4194313, 8978441, 19136521, 40632329, 85983241, 181403657, 381681673, 801112073, 1677721609, 3506438153, 7314866185, 15233712137, 31675383817, 65766686729

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A048494.

[2^(n-1)*(9*n-16)+9 : n in [0..30]]; // Vincenzo Librandi, Sep 25 2011

A048502:=n->2^(n-1)*(9*n-16)+9; seq(A048502(n), n=0..30); # Wesley Ivan Hurt, Dec 04 2013

Table[2^(n-1)*(9n-16)+9, {n,0,30}] (* Wesley Ivan Hurt, Dec 04 2013 *)

Vec((1-3*x+11*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

a(n) = T(8, n), array T given by A048494.
a(n) = 2^(n-1)*(9n-16)+9 = A000079(n-1)*A017185(n-2)+9. - Wesley Ivan Hurt, Dec 04 2013
G.f.: (1-3*x+11*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

Formula and more terms from Ralf Stephan, Jan 15 2004

A274602 Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 5, 2, 3, 11, 9, 3, 4, 19, 20, 13, 4, 5, 29, 35, 29, 17, 5, 6, 41, 54, 51, 38, 21, 6, 7, 55, 77, 79, 67, 47, 25, 7, 8, 71, 104, 113, 104, 83, 56, 29, 8, 9, 89, 135, 153, 149, 129, 99, 65, 33, 9, 10, 109, 170, 199, 202, 185, 154, 115, 74, 37, 10

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2016

Mirrored version of a(n) is T(n,k) = (n-k)*(k+1)^2+k, 0 <= k <= n, read by rows:
0
 1
 5   2
 9  11   3
13  20  19   4
17  29  35  29   5
As an infinite square array (matrix) with comments:
 1   2   3   4   5                      A001477
 5  11  19  29  41                      A028387
 9  20  35  54  77                      A014107
13  29  51  79 113                      A144391
17  38  67 104 149                      A182868
21  47  83 129 185

			0; 1,1; 2,5,2; 3,11,9,3; 4,19,20,13,4; 5,29,35,29,17,5; ...
As an infinite triangular array:
0
1   1
2   5   2
3  11   9    3
4  19  20   13    4
5  29  35   29   17    5
As an infinite square array (matrix) with comments:
0   1   2    3    4    5                   A001477
1   5   9   13   17   21                   A016813
2  11  20   29   38   47                   A017185
3  19  35   51   67   83
4  29  54   79  104  129
5  41  77  113  149  185

Cf. A002064, A001477, A016813, A017185, A062158 (central column). A028387, A014107, A144391, A182868.

Cf. Triangle read by rows: T(n,k) = k*(n-k+1)^m+n-k, 0 <= k <= n: A003056 (m = 0), A059036 (m = 1), A278910 (m = k).

/* As triangle */ [[k*(n-k+1)^2+n-k: k in [0..n]]: n in [0..10]];

Table[k (n - k + 1)^(k + #) + n - k &[2 - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 02 2016 *)

A350521 a(n) = 18*n + 4.

Original entry on oeis.org

4, 22, 40, 58, 76, 94, 112, 130, 148, 166, 184, 202, 220, 238, 256, 274, 292, 310, 328, 346, 364, 382, 400, 418, 436, 454, 472, 490, 508, 526, 544, 562, 580, 598, 616, 634, 652, 670, 688, 706, 724, 742, 760, 778, 796, 814, 832, 850, 868, 886, 904, 922, 940, 958

Offset: 0

Omar E. Pol, Jan 03 2022

Second column of A006370 (the Collatz or 3x+1 map) when it is interpreted as a rectangular array with six columns read by rows.

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

Bisection of A017209.

Cf. A006370, A008600, A016921, A017185, A242215, A298035.

GAP
```
List([0..53], n-> 18*n+4)
```
Magma
```
[18*n+4: n in [0..53]];
```
Maple
```
seq(18*n+4, n=0..53);
```
Mathematica
```
Table[18n+4, {n, 0, 53}]
```
Maxima
```
makelist(18*n+4, n, 0, 53);
```
PARI
```
a(n)=18*n+4
```
Python
```
[18*n+4 for n in range(53)]
```

a(n) = A242215(n) - 1.
a(n) = A298035(n+1) + 1.
From Elmo R. Oliveira, Apr 08 2024: (Start)
G.f.: 2*(2+7*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(2 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A017185(n) = A006370(A016921(n)). (End)

A301621 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 2.

Original entry on oeis.org

11, 29, 47, 83, 101, 119, 137, 173, 191, 209, 227, 263, 281, 299, 317, 353, 371, 389, 407, 443, 461, 479, 497, 533, 551, 569, 587, 623, 641, 659, 677, 713, 731, 749, 767, 803, 821, 839, 857, 893, 911, 929, 947, 983, 1001, 1019, 1037, 1073, 1091, 1109

Offset: 1

Gary Croft, Mar 24 2018

Numbers congruent to 11, 29, 47, or 83 mod 90 with additive sum sequence 11 { + 18 + 18 + 36 + 18} {repeat ...}. Includes all prime numbers greater than 5 with digital root 2.

			11+18=29; 29+18=47; 47+36=83; 83+18=101; 101+18=119.

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

Intersection of A007775 and A017185.

Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=2); # Muniru A Asiru, Apr 22 2018

Flatten[Table[90n - {79, 61, 43, 7}, {n, 30}]] (* Alonso del Arte, Mar 29 2018 *)

Vec(x*(11 + 18*x + 18*x^2 + 36*x^3 + 7*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

n == {11, 29, 47, 83} mod 90.
From Colin Barker, Mar 26 2018: (Start)
G.f.: x*(11 + 18*x + 18*x^2 + 36*x^3 + 7*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

cp's OEIS Frontend

A125758 Numbers congruent to 4 or 7 (mod 9).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A177072 a(n) = (9*n+2)*(9*n+7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A373730 Reduced Collatz function R applied to the numbers 6n+1: a(n) = R(6n+1), where R(k) = (3k+1)/2^r, with r as large as possible.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A048502 a(n) = 2^(n-1)*(9*n-16)+9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A274602 Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A350521 a(n) = 18*n + 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

A177072 a(n) = (9n+2)(9*n+7).

A048502 a(n) = 2^(n-1)(9n-16)+9.