A016993 -id:A016993 - cp's OEIS frontend

A215146 a(n) = 21*n + 1.

1, 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 232, 253, 274, 295, 316, 337, 358, 379, 400, 421, 442, 463, 484, 505, 526, 547, 568, 589, 610, 631, 652, 673, 694, 715, 736, 757, 778, 799, 820, 841, 862, 883, 904, 925, 946, 967, 988, 1009, 1030, 1051, 1072

Offset: 0

Jeremy Gardiner, Aug 04 2012

Jeremy Gardiner, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016777, A016993, A124198, A161709.

I:=[1,22]; [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Apr 19 2018

Range[1, 1000, 21]
LinearRecurrence[{2,-1}, {1,22}, 50] (* G. C. Greubel, Apr 19 2018 *)

for(n=0, 50, print1(21*n + 1, ", ")) \\ G. C. Greubel, Apr 19 2018

From G. C. Greubel, Apr 19 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1+20*x)/(1-x)^2.
E.g.f.: (21*x + 1)*exp(x). (End)
a(n) = A016993(3*n) = A016777(7*n). - Elmo R. Oliveira, Apr 12 2025

A047306 Numbers that are congruent to {0, 2, 3, 4, 5, 6} mod 7.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77

Offset: 1

N. J. A. Sloane

Complement of A016993. - Michel Marcus, Sep 10 2015

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. A016993.

[n: n in [0..100] | n mod 7 in [0] cat [2..6]]; // Vincenzo Librandi, Oct 22 2014

A047306:=n->n+floor((n-2)/6): seq(A047306(n), n=1..100); # Wesley Ivan Hurt, Sep 10 2015

Select[Range[0, 100], MemberQ[{0, 2, 3, 4, 5, 6}, Mod[#, 7]] &] (* Vincenzo Librandi, Oct 22 2014 *)
LinearRecurrence[{1,0,0,0,0,1,-1},{0,2,3,4,5,6,7},70] (* Harvey P. Dale, May 28 2018 *)

concat(0, Vec(x^2*(2+x+x^2+x^3+x^4+x^5)/((1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2) + O(x^30))) \\ Michel Marcus, Oct 22 2014

G.f.: x^2*(2+x+x^2+x^3+x^4+x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Sep 10 2015: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = n + floor((n-2)/6). (End)
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = (42*n-27+3*cos(n*Pi)-12*cos(n*Pi/3)-4*sqrt(3)*sin(2*n*Pi/3))/36.
a(6k) = 7k-1, a(6k-1) = 7k-2, a(6k-2) = 7k-3, a(6k-3) = 7k-4, a(6k-4) = 7k-5, a(6k-5) = 7k-7. (End)

More terms from Michel Marcus, Oct 22 2014

A050478 a(n) = C(n)(8n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 9, 34, 125, 462, 1722, 6468, 24453, 92950, 354926, 1360476, 5231954, 20177164, 78004500, 302211720, 1173076245, 4561139430, 17761336230, 69257611500, 270391268070, 1056823387620

Offset: 0

Barry E. Williams, Dec 24 1999

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Column k=8 of A330965.

Cf. A016993, A000108, A051945.

[Catalan(n)*(8*n+1):n in [0..30]]; // Vincenzo Librandi, Jan 27 2013

R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (7-12*x-7*Sqrt(1-4*x))/(2*x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020

Table[CatalanNumber[n](8n+1),{n,0,20}] (* Harvey P. Dale, May 20 2012 *)

-(n+1)*(8*n-7)*a(n) + 2*(8*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2014
G.f.: (7 - 12*x - 7*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 7*binomial(2*n, n-1) = A000984(n) + 7*A001791(n).
a(n) ~ 2^(2*n+3)/sqrt(Pi*n). (End)

A131873 Right-to-left partial row sums of triangle A131844.

Original entry on oeis.org

1, 8, 4, 15, 8, 7, 22, 12, 11, 10, 29, 16, 15, 14, 13, 36, 20, 19, 18, 17, 16, 43, 24, 23, 22, 21, 20, 19, 50, 28, 27, 26, 25, 24, 23, 22, 57, 32, 31, 30, 29, 28, 27, 26, 25, 64, 36, 35, 34, 33, 32, 31, 30, 29, 28, 71, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 78, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 85, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 92, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40

Offset: 0

Gary W. Adamson, Jul 22 2007

Left column = A016993, 7n + 1: (1, 8, 15, 22, 29, ...).
Right border = 3n + 1: (1, 4, 7, 10, 13, ...).
Row sums = A131874: (1, 12, 30, 55, ...).

			First few rows of the triangle:
   1;
   8,  4;
  15,  8,  7;
  22, 12, 11, 10;
  29, 16, 15, 14, 13;
  36, 20, 19, 18, 17, 16;
  43, 24, 23, 22, 21, 20, 19;
  ...
Row 4 of A131844 is 13, 1, 1, 1, 13, so row 4 of this sequence is 29, 16, 15, 14, 13.

Cf. A131844, A131874, A016993, A131875.

Definition corrected and more terms added by Russ Cox, Apr 18 2024

A168337 a(n) = 1 + 7*floor(n/2).

Original entry on oeis.org

1, 8, 8, 15, 15, 22, 22, 29, 29, 36, 36, 43, 43, 50, 50, 57, 57, 64, 64, 71, 71, 78, 78, 85, 85, 92, 92, 99, 99, 106, 106, 113, 113, 120, 120, 127, 127, 134, 134, 141, 141, 148, 148, 155, 155, 162, 162, 169, 169, 176, 176, 183, 183, 190, 190, 197, 197, 204, 204, 211

Offset: 1

Vincenzo Librandi, Nov 23 2009

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

Cf. A016993, A168333, A168374.

[7*n/2 + 7*(-1)^n/4 - 3/4: n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

Table[ 1 + 7*floor(n/2) , {n,60}] (* Bruno Berselli, Sep 18 2013 *)
CoefficientList[Series[(1 + 7 x - x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 18 2013 *)

a(n) = 7*n - a(n-1) - 5, with n>1, a(1)=1.
From Vincenzo Librandi, Sep 18 2013: (Start)
G.f.: x*(1 + 7*x - x^2)/((1+x)*(x-1)^2).
a(n) = a(n-1) +a(n-2) -a(n-3).
a(n) = (14*n + 7*(-1)^n - 3)/4. (End)
a(n) = A168333(n) - 1 = A168374(n) + 1. - Bruno Berselli, Sep 18 2013
E.g.f.: (1/2)*(-2 + (7*x + 2)*cosh(x) + (7*x - 5)*sinh(x)). - G. C. Greubel, Jul 18 2016

New definition by Bruno Berselli, Sep 18 2013

A249102 Numbers with no 1's in base-7 expansion.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 98, 100, 101, 102, 103, 104, 112, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 128, 129, 130, 131, 132, 133

Offset: 1

Zak Seidov, Oct 21 2014

			14_10 = 20_7, 16_10 = 22_10, 17_10 = 23_7.
14 in base 7 is 20, which contains no 1s, so 14 is in the sequence.
15 in base 7 is 21, which contains one 1, so 15 is not in the sequence.
16 in base 7 is 22, so 16 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Subsequence of A047306. Cf. A023721, A023725, A023729, A023733, A005823. This sequence has no terms in common with A016993.

Select[Range[0, 200], FreeQ[IntegerDigits[#, 7], 1] &] (* Seidov *)
Select[Range[0, 139], DigitCount[#, 7, 1] == 0 &] (* Alonso del Arte, Oct 26 2014 *)

fromdigits(v, b=10)=subst(Pol(v), 'x, b) \\ needed for gp < 2.63 or so
a(n)=a(n)=fromdigits(apply(k->if(k, k+1, 0), digits(n, 6)),7) \\ Charles R Greathouse IV, Oct 30 2014

A291747 Nonprimes of the form 7*k + 1.

Original entry on oeis.org

1, 8, 15, 22, 36, 50, 57, 64, 78, 85, 92, 99, 106, 120, 134, 141, 148, 155, 162, 169, 176, 183, 190, 204, 218, 225, 232, 246, 253, 260, 267, 274, 288, 295, 302, 309, 316, 323, 330, 344, 351, 358, 365, 372, 386, 393, 400, 407, 414, 428, 435, 442, 456, 470, 477, 484, 498

Offset: 1

Vincenzo Librandi, Aug 31 2017

Subsequence of A018252.

Cf. A016993, A091113, A091300, A140444, A291745, A291746.

[n: n in [1..500 by 7] | not IsPrime(n)];

DeleteCases[7 Range[0, 200] + 1, _?PrimeQ]

A354937 Row 7 of A354940: Numbers k for which A345992(k) = 7, divided by 7.

Original entry on oeis.org

4, 5, 8, 11, 15, 19, 22, 25, 29, 32, 39, 43, 47, 50, 53, 57, 61, 64, 67, 71, 78, 81, 89, 92, 95, 99, 103, 106, 109, 113, 127, 131, 134, 137, 141, 151, 155, 162, 169, 173, 176, 179, 183, 190, 193, 197, 211, 218, 229, 232, 239, 243, 256, 257, 263, 267, 271, 274, 277, 281, 291, 295, 302, 309, 313, 316, 323, 337, 344

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Apparently, all terms are either of the form 7k+1 (in A016993), 7k+4 (A017029) or 7k+5 (A017041).

Row 7 of A354940.

Cf. A016993, A017029, A017041, A344005, A345992.

q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 7*n], m++]; GCD[7*n, m] == 7]; Select[Range[345], q] (* Amiram Eldar, Jun 17 2022 *)

PARI
```
A354937(n) = A354940sq(7,n);
```

A001526 a(n) = (7n+1)(7*n+6).

Original entry on oeis.org

6, 104, 300, 594, 986, 1476, 2064, 2750, 3534, 4416, 5396, 6474, 7650, 8924, 10296, 11766, 13334, 15000, 16764, 18626, 20586, 22644, 24800, 27054, 29406, 31856, 34404, 37050, 39794, 42636, 45576, 48614, 51750, 54984, 58316, 61746, 65274, 68900, 72624, 76446

Offset: 0

N. J. A. Sloane

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

Cf. A016993, A017053.

a[n_] := (7*n + 1)*(7*n + 6); Array[a, 40, 0] (* Amiram Eldar, Feb 19 2023 *)

a(n)=(7*n+1)*(7*n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 98*n + a(n-1) with a(0)=6. - Vincenzo Librandi, Nov 12 2010
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A016993(n)*A017053(n).
Sum_{n>=0} 1/a(n) = cot(Pi/7)*Pi/35 = 0.186388....
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/7)*cos(sqrt(29)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/7)*cos(sqrt(3/7)*Pi/2). (End)
G.f.: -2*(3+43*x+3*x^2)/(x-1)^3. - R. J. Mathar, Apr 23 2024
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(6 + 49*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A137183 Lucky numbers (A000959) which are congruent to 1 mod 7.

Original entry on oeis.org

1, 15, 43, 99, 127, 141, 169, 211, 267, 393, 421, 463, 477, 519, 631, 645, 673, 729, 841, 855, 883, 897, 925, 981, 1009, 1023, 1093, 1107, 1219, 1233, 1261, 1275, 1303, 1387, 1401, 1471, 1485, 1597, 1611, 1639, 1723, 1737, 1765, 1933, 2031, 2059, 2115, 2283

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A016993.

cp's OEIS Frontend

A215146 a(n) = 21*n + 1.

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A047306 Numbers that are congruent to {0, 2, 3, 4, 5, 6} mod 7.

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A050478 a(n) = C(n)*(8*n+1) where C(n) = Catalan numbers (A000108).

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A131873 Right-to-left partial row sums of triangle A131844.

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A168337 a(n) = 1 + 7*floor(n/2).

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A249102 Numbers with no 1's in base-7 expansion.

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A291747 Nonprimes of the form 7*k + 1.

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Magma

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A354937 Row 7 of A354940: Numbers k for which A345992(k) = 7, divided by 7.

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A050478 a(n) = C(n)(8n+1) where C(n) = Catalan numbers (A000108).

A001526 a(n) = (7n+1)(7*n+6).