A017005 -id:A017005 - cp's OEIS frontend

A017089 a(n) = 8*n + 2.

2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 226, 234, 242, 250, 258, 266, 274, 282, 290, 298, 306, 314, 322, 330, 338, 346, 354, 362, 370, 378, 386, 394, 402, 410, 418, 426

Offset: 0

N. J. A. Sloane, Dec 11 1996

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 33 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 81 ).
First differences of A002939. - Aaron David Fairbanks, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002939, A008589, A008590, A016813, A016993, A017005, A017017, A017077.

a017089 = (+ 2) . (* 8)
a017089_list = [2, 10 ..]  -- Reinhard Zumkeller, Jun 07 2015

[8*n+2: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017089:=n->8*n+2; seq(A017089(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Range[2, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n) = 8*n+2; \\ Michel Marcus, Sep 17 2015

a(n) = 8*n+2; a(n) = 2*a(n-1)-a(n-2). - Vincenzo Librandi, May 28 2011
Sum_{n>=0} (-1)^n/a(n) = (Pi + 2*log(cot(Pi/8)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
From Elmo R. Oliveira, Mar 17 2024: (Start)
G.f.: 2*(1+3*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(1 + 4*x).
a(n) = 2*A016813(n) = A008590(n) + 2. (End)

A017029 a(n) = 7*n + 4.

Original entry on oeis.org

4, 11, 18, 25, 32, 39, 46, 53, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 130, 137, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 214, 221, 228, 235, 242, 249, 256, 263, 270, 277, 284, 291, 298, 305, 312, 319, 326, 333, 340, 347, 354, 361, 368, 375, 382

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005, A017017.

Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.

[7*n + 4: n in [0..60]]; // Vincenzo Librandi, Jun 18 2011

Range[4,1000,7] (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)
CoefficientList[Series[(3*x + 4)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 27 2013 *)
LinearRecurrence[{2,-1},{4,11},60] (* Harvey P. Dale, Mar 27 2025 *)

a(n)=7*n+4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: (3*x + 4)/(1-x)^2. - Vincenzo Librandi, Jan 27 2013
From Elmo R. Oliveira, Apr 12 2025: (Start)
E.g.f.: exp(x)*(4 + 7*x).
a(n) = 2*a(n-1) - a(n-2). (End)

Extended by Ray Chandler, Jan 25 2005

A017017 a(n) = 7*n + 3.

Original entry on oeis.org

3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 199, 206, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311, 318, 325, 332, 339, 346, 353, 360, 367, 374, 381

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005.

[7*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017017:=n->7*n + 3; seq(A017017(n), n=0..100); # Wesley Ivan Hurt, Feb 24 2014

Range[3, 600, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n)=7*n+3 \\ Charles R Greathouse IV, Jul 02 2013

[7*n+3 for n in range(80)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (3 + 4*x)/(1 - x)^2.
E.g.f.: (3 + 7*x)*exp(x). (End)

A017053 a(n) = 7*n + 6.

Original entry on oeis.org

6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 160, 167, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 251, 258, 265, 272, 279, 286, 293, 300, 307, 314, 321, 328, 335, 342, 349, 356, 363, 370, 377, 384

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 955
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005, A017017.

[7*n + 6: n in [0..60]]; // Vincenzo Librandi, Jun 18 2011

Array[7*#+6&,100,0] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
LinearRecurrence[{2,-1},{6,13},60] (* Harvey P. Dale, Apr 13 2022 *)

a(n)=7*n+6 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Mar 17 2023

G.f.: (6+x)/(1-x)^2. - Wesley Ivan Hurt, Dec 28 2023

A198017 a(n) = n(7n + 11)/2 + 1.

Original entry on oeis.org

1, 10, 26, 49, 79, 116, 160, 211, 269, 334, 406, 485, 571, 664, 764, 871, 985, 1106, 1234, 1369, 1511, 1660, 1816, 1979, 2149, 2326, 2510, 2701, 2899, 3104, 3316, 3535, 3761, 3994, 4234, 4481, 4735, 4996, 5264, 5539, 5821, 6110, 6406, 6709, 7019, 7336, 7660, 7991

Offset: 0

Bruno Berselli, Oct 21 2011 - based on remarks and sequences by Omar E. Pol

First bisection of A193053 (see also the numerical spiral illustrated in the Links section).

The inverse binomial transform yields 1, 9, 7, 0, 0 (0 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A195020 (vertices of the numerical spiral in link).
Cf. A017005 (first differences).
Cf. A069099, A001106.
Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A193053, A195021, A195023-A195030, A195320 [incomplete list].

Magma
```
[n*(7*n+11)/2+1: n in [0..47]];
```

Table[(n(7n+11))/2+1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{1,10,26},60] (* Harvey P. Dale, Mar 03 2013 *)

for(n=0, 47, print1(n*(7*n+11)/2+1", "));

G.f.: (1 + 7*x - x^2)/(1-x)^3.
a(n) = A195020(2*n) + 2*n + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = 2*a(n-1) - a(n-2) + 7.
From Elmo R. Oliveira, Dec 24 2024: (Start)
E.g.f.: exp(x)*(2 + 18*x + 7*x^2)/2.
a(n) = n + A001106(n+1). (End)

A134500 a(n) = Fibonacci(7n + 2).

Original entry on oeis.org

1, 34, 987, 28657, 832040, 24157817, 701408733, 20365011074, 591286729879, 17167680177565, 498454011879264, 14472334024676221, 420196140727489673, 12200160415121876738, 354224848179261915075, 10284720757613717413913

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1)

Cf. A000045, A134498, A134499, A134500, A134501, A134502, A134503, A134504.

[Fibonacci(7*n +2): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[7n+2], {n, 0, 30}]
```

a(n)=fibonacci(7*n+2) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: (-1-5*x) / (-1 + 29*x + x^2). - R. J. Mathar, Apr 17 2011

a(n) = A000045(A017005(n)). - Michel Marcus, Nov 07 2013

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A342758 Array read by ascending antidiagonals: T(k, n) is the maximum value of the magic constant in a perimeter-magic k-gon of order n.

Original entry on oeis.org

12, 15, 23, 19, 30, 37, 22, 37, 48, 54, 26, 44, 60, 71, 74, 29, 51, 71, 88, 97, 97, 33, 58, 83, 105, 121, 128, 123, 36, 65, 94, 122, 144, 159, 162, 152, 40, 72, 106, 139, 168, 190, 202, 201, 184, 43, 79, 117, 156, 191, 221, 241, 250, 243, 219, 47, 86, 129, 173, 215, 252, 281, 299, 303, 290, 257

Offset: 3

Stefano Spezia, Mar 21 2021

			The array begins:
k\n|  3   4   5    6    7 ...
---+---------------------
3  | 12  23  37   54   74 ...
4  | 15  30  48   71   97 ...
5  | 19  37  60   88  121 ...
6  | 22  44  71  105  144 ...
7  | 26  51  83  122  168 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 11 and 13).

Cf. A017005 (n = 4), A135503 (diagonal), A341740 (k = 3), A342719, A342757 (minimum).

T[k_,n_]:= (n+k(n^2-2)+(Mod[k,2]-1)Mod[n,2])/2; Table[T[k+3-n,n],{k,3,13},{n,3,k}]//Flatten

G.f.: (- x^2*(2*y^2 + y - 1) - x*(y^2 + 2*y - 1) + (y - 1)*y^2)/((x - 1)^2*(x + 1)*(y - 1)^3*(y + 1)).
T(k, n) = (n^2/2 - 1)*k + n/2 if n is even or both n and k are odd.
T(k, n) = (n^2/2 - 1)*k + (n - 1)/2 if n is odd and k is even.
T(k, n) = (n + k*(n^2 - 2) + ((k mod 2) - 1)*(n mod 2))/2.

A139608 a(n) = 28*n + 8.

Original entry on oeis.org

8, 36, 64, 92, 120, 148, 176, 204, 232, 260, 288, 316, 344, 372, 400, 428, 456, 484, 512, 540, 568, 596, 624, 652, 680, 708, 736, 764, 792, 820, 848, 876, 904, 932, 960, 988, 1016, 1044, 1072, 1100, 1128, 1156, 1184, 1212, 1240, 1268, 1296, 1324, 1352, 1380

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 8th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017005, A057145, A135628, A139600, A316466.

[4*(7*n + 2): n in [0..50]]; // Vincenzo Librandi, Jul 13 2011

Range[8, 7000, 28] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2,-1},{8,36},50] (* or *) NestList[28+#&,8,50] (* Harvey P. Dale, Dec 14 2012 *)

a(n)=28*n+8 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,8).
a(n) = 2*a(n-1) - a(n-2); a(0)=8, a(1)=36. - Harvey P. Dale, Dec 14 2012
G.f.: 4*(2+5*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 4*exp(x)*(2 + 7*x).
a(n) = 4*A017005(n) = A135628(n) + 8 = A316466(n+1) - A316466(n). (End)

A153384 Numbers n such that 24*n+1 is not prime.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 9, 11, 12, 15, 16, 20, 21, 22, 23, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 44, 45, 46, 49, 51, 53, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 68, 70, 71, 72, 76, 77, 79, 80, 81, 82, 85, 86, 91, 92, 93, 94, 96, 97, 98, 100, 101, 102

Offset: 1

Vincenzo Librandi, Dec 25 2008

Contains all numbers == 1 (mod 5), ==2 (mod 7), ==5 (mod 11), == 7 (mod 13), == 12 (mod 17), == 15 (mod 19), == 22 (mod 23), == 6 (mod 29) etc, so it is the union of A016861, A017005, A017449, A269044, etc. - R. J. Mathar, Jun 10 2020

Even terms of A153383, halved. - R. J. Mathar, Jun 10 2020

			Triangle begins:
*;
*,1;
*,*,2;
*,*,*,*;
*,*,*,*,5;
*,*,*,*,*,7;
*,*,*,*,*,*,*;
*,*,*,*,*,*,*,12;
*,*,*,*,*,*,*,*,15;
*,*,*,*,*,*,*,*,*,*;
*,*,*,*,*,*,*,*,*,*,22; etc.
where * marks the non-integer values of (2*h*k + k + h)/12 with h >= k >= 1. - _Vincenzo Librandi_, Jan 14 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001318, A111174 (complement).

[n: n in [0..150] | not IsPrime(24*n + 1)]; // Vincenzo Librandi, Jan 14 2013

Select[Range[0, 200], !PrimeQ[24 # + 1] &] (* Vincenzo Librandi, Jan 14 2013 *)

0 added by Arkadiusz Wesolowski, Aug 03 2011

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.

Original entry on oeis.org

10, 13, 18, 16, 23, 30, 19, 28, 37, 46, 22, 33, 44, 55, 66, 25, 38, 51, 64, 77, 90, 28, 43, 58, 73, 88, 103, 118, 31, 48, 65, 82, 99, 116, 133, 150, 34, 53, 72, 91, 110, 129, 148, 167, 186, 37, 58, 79, 100, 121, 142, 163, 184, 205, 226, 40, 63, 86, 109, 132, 155, 178

Offset: 1

Vincenzo Librandi, Aug 02 2009

The numbers 2*T(n,m)-11 = (2*n+1)*(2*m+1) are not prime, and 2*T(n,n) = (2n+1)^2.

First column: A112414, second column: A016885, third column: A017005, fourth column: A017173. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  10;
  13, 18;
  16, 23, 30;
  19, 28, 37, 46;
  22, 33, 44, 55,  66;
  25, 38, 51, 64,  77,  90;
  28, 43, 58, 73,  88,  103, 118;
  31, 48, 65, 82,  99,  116, 133, 150;
  34, 53, 72, 91,  110, 129, 148, 167, 186;
  37, 58, 79, 100, 121, 142, 163, 184, 205, 226;
  40, 63, 86, 109, 132, 155, 178, 201, 224, 247, 270;
  etc.

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

Cf. A153045, A154685, A163657, A112414, A016885, A017005, A017173.

[2*n*k + n + k + 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k +  6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

T(n,m) = A154685(n,m)+2 = A163657(n,m)-2. [R. J. Mathar, Oct 22 2009]

Comment clarified by R. J. Mathar, Oct 22 2009

cp's OEIS Frontend

A017089 a(n) = 8*n + 2.

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A017029 a(n) = 7*n + 4.

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A017017 a(n) = 7*n + 3.

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A017053 a(n) = 7*n + 6.

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A198017 a(n) = n*(7*n + 11)/2 + 1.

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A134500 a(n) = Fibonacci(7n + 2).

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A342758 Array read by ascending antidiagonals: T(k, n) is the maximum value of the magic constant in a perimeter-magic k-gon of order n.

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A198017 a(n) = n(7n + 11)/2 + 1.

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.