A017029 -id:A017029 - cp's OEIS frontend

A100776 a(n) = 997*n + 1009.

1009, 2006, 3003, 4000, 4997, 5994, 6991, 7988, 8985, 9982, 10979, 11976, 12973, 13970, 14967, 15964, 16961, 17958, 18955, 19952, 20949, 21946, 22943, 23940, 24937, 25934, 26931, 27928, 28925, 29922, 30919, 31916, 32913, 33910, 34907, 35904, 36901, 37898, 38895

Offset: 0

Parthasarathy Nambi, Jan 03 2005

Note that 997 is the largest three-digit prime and 1009 is the smallest four-digit prime.

			If n=1, 997*1 + 1009 = 2006.
If n=2, 997*2 + 1009 = 3003.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017029, A100775, A101086, A101442.

[997 * n + 1009: n in [0..50]]; // Vincenzo Librandi, Jul 14 2011

A100776:=n->997 * n + 1009; seq(A100776(n), n=0..50); # Wesley Ivan Hurt, Jan 22 2014

Table[997 n + 1009, {n, 0, 50}] (* Wesley Ivan Hurt, Jan 22 2014 *)

a(n)=997*n+1009 \\ Charles R Greathouse IV, Jul 10 2016

From Elmo R. Oliveira, Dec 07 2024: (Start)
G.f.: (1009 - 12*x)/(1 - x)^2.
E.g.f.: (1009 + 997*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

Edited by Ray Chandler, Jan 25 2005

A101442 a(n) = 9973*n + 10007.

Original entry on oeis.org

10007, 19980, 29953, 39926, 49899, 59872, 69845, 79818, 89791, 99764, 109737, 119710, 129683, 139656, 149629, 159602, 169575, 179548, 189521, 199494, 209467, 219440, 229413, 239386, 249359, 259332, 269305, 279278, 289251, 299224, 309197, 319170, 329143, 339116

Offset: 0

Parthasarathy Nambi, Jan 18 2005

9973 is the largest four-digit prime and 10007 is the smallest five-digit prime.

			If n=14, then 9973*14 + 10007 = 149629 (a prime).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Chris Caldwell, The first 100,008 primes.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017029, A100775, A100776, A101444.

[9973*n + 10007: n in [0..50]]; // Vincenzo Librandi, Jul 14 2011

9973*Range[0,30]+10007 (* or *) LinearRecurrence[{2,-1},{10007,19980},40] (* Harvey P. Dale, Sep 02 2015 *)

a(n)=9973*n+10007 \\ Charles R Greathouse IV, Oct 16 2015

From Elmo R. Oliveira, Dec 07 2024: (Start)
G.f.: (10007 - 34*x)/(1 - x)^2.
E.g.f.: (10007 + 9973*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)

Extended by Ray Chandler, Jan 25 2005

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

Original entry on oeis.org

10, 16, 26, 22, 36, 50, 28, 46, 64, 82, 34, 56, 78, 100, 122, 40, 66, 92, 118, 144, 170, 46, 76, 106, 136, 166, 196, 226, 52, 86, 120, 154, 188, 222, 256, 290, 58, 96, 134, 172, 210, 248, 286, 324, 362, 64, 106, 148, 190, 232, 274, 316, 358, 400, 442, 70, 116, 162

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016957, second column: A017341, third column: 2*A017029, fourth column: A082286. - Vincenzo Librandi, Nov 21 2012
Conjecture: Let p = prime number. If 2^p belongs to the sequence, then 2^p-1 is not a Mersenne prime. - Vincenzo Librandi, Dec 12 2012
Conjecture is true because if T(n, k) = 2^p with p prime, then 2^p-1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p-1 is not prime. - Michel Marcus, May 31 2015
It appears that T(m,p) = 2^p for Lucasian primes (A002515) greater than 3. For instance: T(44, 11) = 2^11, T(89240, 23) = 2^23. - Michel Marcus, May 28 2015
For n > 1, ascending numbers along the diagonal are also terms of the even principal diagonal of a 2n X 2n spiral (A137928). - Avi Friedlich, May 21 2015

			Triangle begins
  10;
  16,  26;
  22,  36,  50;
  28,  46,  64,  82;
  34,  56,  78, 100, 122;
  40,  66,  92, 118, 144, 170;
  46,  76, 106, 136, 166, 196, 226;
  52,  86, 120, 154, 188, 222, 256, 290;
  58,  96, 134, 172, 210, 248, 286, 324, 362;
  64, 106, 148, 190, 232, 274, 316, 358, 400, 442;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.

[4*n*k + 2*n + 2*k + 2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

seq(seq( 2*(2*n*k+n+k+1), k=1..n), n=1..15) # G. C. Greubel, Mar 21 2021

T[n_,k_]:=4*n*k + 2*n + 2*k + 2; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k+n+k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 21 2021

T(n, k) = 2*A144650(n, k).

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n + 3) = n*A014105(n+2) =

Edited by Robert Hochberg, Jun 21 2010

A155550 Triangle read by rows where T(m,n)=2mn + m + n - 6.

Original entry on oeis.org

-2, 1, 6, 4, 11, 18, 7, 16, 25, 34, 10, 21, 32, 43, 54, 13, 26, 39, 52, 65, 78, 16, 31, 46, 61, 76, 91, 106, 19, 36, 53, 70, 87, 104, 121, 138, 22, 41, 60, 79, 98, 117, 136, 155, 174, 25, 46, 67, 88, 109, 130, 151, 172, 193, 214, 28, 51, 74, 97, 120, 143, 166, 189, 212

Offset: 1

Vincenzo Librandi, Jan 24 2009

Numbers n such that 2n+13 is not prime.

First column: A016777, second column: A016861, third column: A017029, fourth column: A017245. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
-2;
1,  6;
4,  11, 18;
7,  16, 25, 34;
10, 21, 32, 43, 54;
13, 26, 39, 52, 65,  78;
16, 31, 46, 61, 76,  91,  106;
19, 36, 53, 70, 87,  104, 121, 138;
22, 41, 60, 79, 98,  117, 136, 155, 174;
25, 46, 67, 88, 109, 130, 151, 172, 193, 214; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153082, A016777, A016861, A017029, A017245.

[2*n*k + n + k - 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

t[n_,k_]:= 2 n*k + n + k - 6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

A177071 a(n) = (7n + 3)(7*n + 4).

Original entry on oeis.org

12, 110, 306, 600, 992, 1482, 2070, 2756, 3540, 4422, 5402, 6480, 7656, 8930, 10302, 11772, 13340, 15006, 16770, 18632, 20592, 22650, 24806, 27060, 29412, 31862, 34410, 37056, 39800, 42642, 45582, 48620, 51756, 54990, 58322, 61752, 65280, 68906, 72630, 76452

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. Zumkeller's contribution in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2, therefore a(n) = 49*A002061(n+1) - 37. - Bruno Berselli, Aug 24 2010

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

Cf. A002061, A017017, A017029, A061792, A177059.

Table[(7n+3)(7n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{12,110,306},40] (* Harvey P. Dale, Oct 09 2011 *)

a(n)=2*binomial(7*n+4,2) \\ Charles R Greathouse IV, Jan 11 2012

a(n) = 98*n + a(n-1) with n > 0, a(0)=12.
From Harvey P. Dale, Oct 09 2011: (Start)
a(0)=12, a(1)=110, a(2)=306, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -((2*(x+6)*(6*x+1))/(x-1)^3). (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017017(n)*A017029(n).
Sum_{n>=0} 1/a(n) = tan(Pi/14)*Pi/7.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/14)*cos(sqrt(5)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/14)*cosh(sqrt(3)*Pi/14). (End)
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(12 + 49*x*(2 + x)).
a(n) = 2*A061792(n). (End)

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

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

A017030 a(n) = (7*n + 4)^2.

Original entry on oeis.org

16, 121, 324, 625, 1024, 1521, 2116, 2809, 3600, 4489, 5476, 6561, 7744, 9025, 10404, 11881, 13456, 15129, 16900, 18769, 20736, 22801, 24964, 27225, 29584, 32041, 34596, 37249, 40000, 42849, 45796

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[(7*n+4)^2: n in [0..40] ]; // Vincenzo Librandi, Jul 16 2011

A017030:=n->(7*n+4)^2; seq(A017030(n), n=0..100); # Wesley Ivan Hurt, Nov 11 2013

Table[(7n+4)^2, {n,0,100}] (* Wesley Ivan Hurt, Nov 11 2013 *)
LinearRecurrence[{3,-3,1},{16,121,324},40] (* Harvey P. Dale, May 25 2019 *)

a(n)=(7*n+4)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A017029(n)^2. - Michel Marcus, Nov 11 2013

Sum{n>=0} 1/a(n) = psi'(4/7)/49 = 0.080438723... - R. J. Mathar, May 07 2024

A047374 Numbers that are congruent to {4, 5} mod 7.

Original entry on oeis.org

4, 5, 11, 12, 18, 19, 25, 26, 32, 33, 39, 40, 46, 47, 53, 54, 60, 61, 67, 68, 74, 75, 81, 82, 88, 89, 95, 96, 102, 103, 109, 110, 116, 117, 123, 124, 130, 131, 137, 138, 144, 145, 151, 152, 158, 159, 165, 166, 172

Offset: 1

N. J. A. Sloane

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

Cf. A017029, A017041.

Select[Range[200], MemberQ[{4, 5}, Mod[#, 7]] &] (* Amiram Eldar, May 07 2021 *)

a(n) = (14*n - 5*(-1)^n - 3)/4 \\ David Lovler, Sep 15 2022

G.f.: x*(4 + x + 2*x^2)/((1 + x)*(x - 1)^2). - R. J. Mathar, Dec 04 2011
a(n) = -(5/4)*(-1)^n + 7*(n-1)/2 + 11/4. - Viet Quoc Le Tran, Jun 14 2014
a(n) = (14*n - 5*(-1)^n - 3)/4. - David Lovler, Sep 15 2022
E.g.f.: 2 + ((14*x - 3)*exp(x) - 5*exp(-x))/4. - David Lovler, Sep 15 2022

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 65, 67, 68, 71, 73, 74, 77, 78, 80, 81, 83, 86, 88, 89, 92, 93, 95, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 116, 118, 119, 122

Offset: 1

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

n is in the sequence iff n is in A016789 or in A016885 or in A017029.
First differences are periodic with period length 57. Least common multiple of 3, 5, 7 is 105; number of terms <= 105 is 57.
Sequence is not essentially the same as A053726: a(n) = A053726(n-3) for 3 < n < 33, a(34)=62, A053726(34-3)=61.
Sequence is not essentially the same as A104275: a(n) = A104275(n-2) for 3 < n < 33, a(34)=62, A104275(34-3)=61.

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

Cf. A016789, A016885, A017029, A053726, A104275.

IsA186041:=func< n | exists{ k: k in [0..n div 3] | n in [3*k+2, 5*k+3, 7*k+4] } >; [ n: n in [1..200] | IsA186041(n) ];

Take[With[{no=50},Union[Join[3Range[0,no]+2,5Range[0,no]+3,7Range[0,no]+4]]],70]  (* Harvey P. Dale, Feb 16 2011 *)

a(n) = a(n-57) + 105.
a(n) = a(n-1) + a(n-57) - a(n-58).
G.f.: x*(x^57 + x^56 + x^55 + x^54 + 3*x^53 + 3*x^52 + 2*x^51 + x^50 + 3*x^49 + x^48 + 2*x^47 + 3*x^46 + 2*x^45 + x^44 + 2*x^43 + x^42 + 3*x^41 + x^40 + 2*x^39 + 3*x^38 + x^37 + 2*x^36 + 2*x^35 + x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 3*x^29 + 3*x^28 + 2*x^27 + x^26 + x^25 + 2*x^24 + x^23 + 2*x^22 + 2*x^21 + x^20 + 3*x^19 + 2*x^18 + x^17 + 3*x^16 + x^15 + 2*x^14 + x^13 + 2*x^12 + 3*x^11 + 2*x^10 + x^9 + 3*x^8 + x^7 + 2*x^6 + 3*x^5 + 3*x^4 + x^3 + x^2 + x + 2) / ((x - 1)^2*(x^2 + x + 1)*(x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)).

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);
```

cp's OEIS Frontend

A100776 a(n) = 997*n + 1009.

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A101442 a(n) = 9973*n + 10007.

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A155151 Triangle T(n, k) = 4*n*k + 2*n + 2*k + 2, read by rows.

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A155550 Triangle read by rows where T(m,n)=2*m*n + m + n - 6.

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A177071 a(n) = (7*n + 3)*(7*n + 4).

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2*n - 2*k*n + 2*n^2, with 0 <= k < n.

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A017030 a(n) = (7*n + 4)^2.

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

A155550 Triangle read by rows where T(m,n)=2mn + m + n - 6.

A177071 a(n) = (7n + 3)(7*n + 4).

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.