A017041 -id:A017041 - cp's OEIS frontend

A269100 a(n) = 13*n + 11.

11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739

Offset: 0

Bruno Berselli, Feb 19 2016

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k =  2: A153080,
k =  6: A186113,
k =  7: A269044,
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A094784, A106389.
Cf. A140373.
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), this sequence (q=11).

Magma
```
[13*n+11: n in [0..60]];
```

13 Range[0,60] + 11
Range[11, 800, 13]
Table[13 n + 11, {n, 0, 60}] (* Bruno Berselli, Feb 22 2016 *)
LinearRecurrence[{2,-1},{11,24},60] (* Harvey P. Dale, Jun 14 2023 *)

Maxima
```
makelist(13*n+11, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+11)
```
Python
```
[13*n+11 for n in range(61)]
```
Sage
```
[13*n+11 for n in range(61)]
```

G.f.: (11 + 2*x)/(1 - x)^2.
a(n) = -A153080(-n-1).
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
E.g.f.: (11 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A125199 Triangle read by rows: T(n,k) = 4nk - n - k, 1<=k<=n.

Original entry on oeis.org

2, 5, 12, 8, 19, 30, 11, 26, 41, 56, 14, 33, 52, 71, 90, 17, 40, 63, 86, 109, 132, 20, 47, 74, 101, 128, 155, 182, 23, 54, 85, 116, 147, 178, 209, 240, 26, 61, 96, 131, 166, 201, 236, 271, 306, 29, 68, 107, 146, 185, 224, 263, 302, 341, 380, 32, 75, 118, 161, 204, 247

Offset: 1

Reinhard Zumkeller, Nov 24 2006

A124934 gives the range: for n,k with 1<=k<=n exists at least one m such that A124934(m)=T(n,k);
row sums give A125200; central terms give A125201;
T(n,1) = A016789(n-1);
T(n,2) = A017041(n-1) for n>1;
T(n,3) = A017485(n-1) for n>2;
T(n,n-1) = A125202(n) for n>1;
T(n,n) = A002939(n).

Flatten[Table[4*n*k-n-k,{n,15},{k,n}]] (* Harvey P. Dale, Nov 15 2014 *)

A047382 Numbers that are congruent to {0, 5} mod 7.

Original entry on oeis.org

0, 5, 7, 12, 14, 19, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 61, 63, 68, 70, 75, 77, 82, 84, 89, 91, 96, 98, 103, 105, 110, 112, 117, 119, 124, 126, 131, 133, 138, 140, 145, 147, 152, 154, 159, 161, 166, 168

Offset: 1

N. J. A. Sloane

Except for the first term, numbers m such that 36*m^2 + 72*m + 35 = (6*m+5)*(6*m+7) is not of the form p*(p+2), with p prime. - Vincenzo Librandi, Aug 05 2010

Nonnegative k such that k or 4*k + 1 is divisible by 7. - Bruno Berselli, Feb 13 2018

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

Cf. A008589, A017041.

&cat[[7*n,7*n+5]: n in [0..23]];  // Bruno Berselli, Oct 17 2011

{#, 5 + #} &/@ (7 Range[0, 30]) // Flatten (* or *) LinearRecurrence[{1, 1, -1}, {0, 5, 7}, 60] (* Harvey P. Dale, Dec 01 2016 *)

a(n) = (14*n + 3*(-1)^n - 11)/4 \\ David Lovler, Sep 11 2022

a(n) = 7*n - a(n-1) - 9 for n>1, with a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=A005009(k-1)=7*2^(k-1) for k>0. - Philippe Deléham, Oct 17 2011
From Bruno Berselli, Oct 17 2011:  (Start)
G.f.: x^2*(5 + 2*x)/((1 + x)*(1 - x)^2).
a(n) = (14*n + 3*(-1)^n - 11)/4.
a(-n) = -A047352(n+2). (End)
a(n) = ceiling((7/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
E.g.f.: 2 + ((14*x - 11)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 11 2022

A126980 a(n) = 14*n + 47.

Original entry on oeis.org

47, 61, 75, 89, 103, 117, 131, 145, 159, 173, 187, 201, 215, 229, 243, 257, 271, 285, 299, 313, 327, 341, 355, 369, 383, 397, 411, 425, 439, 453, 467, 481, 495, 509, 523, 537, 551, 565, 579, 593, 607, 621, 635, 649, 663, 677, 691, 705, 719, 733, 747

Offset: 0

Robert H Barbour, Mar 20 2007, Jun 12 2007

Superhighway created by 'LQTL Ant' L90R135L90R135 from iteration 47 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the turn angle (in degrees) at each iteration.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, 2000.
S. Wolfram, A New Kind of Science, 1st ed. Il.: Wolfram Media Inc., 2002.

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

Cf. A017041, A019430, A094867, A126978, A126979.

[14*n + 47: n in [0..60] ]; // Vincenzo Librandi, Jul 18 2011

Table[14*n + 47, {n, 0, 60}] (* Stefan Steinerberger, Jun 17 2007 *)
LinearRecurrence[{2,-1}, {47,61}, 50] (* G. C. Greubel, May 30 2016 *)

From Chai Wah Wu, May 30 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (-33*x + 47)/(x - 1)^2. (End)
E.g.f.: (47 + 14*x)*exp(x). - G. C. Greubel, May 30 2016

More terms from Stefan Steinerberger and Diana L. Mecum, Jun 17 2007

A155546 Triangle read by rows where T(m,n)=2mn + m + n - 5, 1<=n<=m.

Original entry on oeis.org

-1, 2, 7, 5, 12, 19, 8, 17, 26, 35, 11, 22, 33, 44, 55, 14, 27, 40, 53, 66, 79, 17, 32, 47, 62, 77, 92, 107, 20, 37, 54, 71, 88, 105, 122, 139, 23, 42, 61, 80, 99, 118, 137, 156, 175, 26, 47, 68, 89, 110, 131, 152, 173, 194, 215, 29, 52, 75, 98, 121, 144, 167, 190, 213

Offset: 1

Vincenzo Librandi, Jan 24 2009

2*T(m,n)+11 = (2*m+1)*(2*n+1) is not prime.

First column: A016789, second column: A016873, third column: A017041, fourth column: A017257. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
-1;
2,  7;
5,  12, 19;
8,  17, 26, 35;
11, 22, 33, 44, 55;
14, 27, 40, 53, 66,  79;
17, 32, 47, 62, 77,  92,  107;
20, 37, 54, 71, 88,  105, 122, 139;
23, 42, 61, 80, 99,  118, 137, 156, 175;
26, 47, 68, 89, 110, 131, 152, 173, 194, 215; etc.

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

Cf. A101448, A153083, A016789, A016873, A017041, A017257.

[2*n*k + n + k - 5: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

t[n_,k_]:= 2 n*k + n + k - 5; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

A163674 Triangle T(n,m) = 2mn + m + n + 9 read by rows.

Original entry on oeis.org

13, 16, 21, 19, 26, 33, 22, 31, 40, 49, 25, 36, 47, 58, 69, 28, 41, 54, 67, 80, 93, 31, 46, 61, 76, 91, 106, 121, 34, 51, 68, 85, 102, 119, 136, 153, 37, 56, 75, 94, 113, 132, 151, 170, 189, 40, 61, 82, 103, 124, 145, 166, 187, 208, 229, 43, 66, 89, 112, 135, 158, 181

Offset: 1

Vincenzo Librandi, Aug 03 2009

2*T(m,n) - 17 =(2*n+1)*(2*m+1) and 2*T(n,n) - 17 is a square. Also:
first column:  A112414;
second column: A016861;
third column:  A017041;
fourth column: A017209. [Vincenzo Librandi, Nov 20 2012]

			Triangle begins:
  13;
  16,  21;
  19,  26,  33;
  22,  31,  40,  49;
  25,  36,  47,  58,  69;
  28,  41,  54,  67,  80,  93;
  31,  46,  61,  76,  91, 106, 121;
  34,  51,  68,  85, 102, 119, 136, 153;

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

Cf. A016861, A017041, A017209, A112414, A153051, A163657.

[2*n*k + n + k + 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

for(n=1,10, for(m=1,n, print1(2*m*n + m + n + 9, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A163657(n,m) + 1.

Edited by R. J. Mathar, Oct 12 2009

A155705 Triangle read by rows where T(m,n) = 2mn + m + n + 2.

Original entry on oeis.org

6, 9, 14, 12, 19, 26, 15, 24, 33, 42, 18, 29, 40, 51, 62, 21, 34, 47, 60, 73, 86, 24, 39, 54, 69, 84, 99, 114, 27, 44, 61, 78, 95, 112, 129, 146, 30, 49, 68, 87, 106, 125, 144, 163, 182, 33, 54, 75, 96, 117, 138, 159, 180, 201, 222, 36, 59, 82, 105, 128, 151

Offset: 1

Vincenzo Librandi, Jan 25 2009

2*T(m,n)-3 = (2*m+1)*(2*n+1) is not prime, obviously. Also: first column: 3*A020725; second column: A016897; third column: A017041; fourth column: 3*A016789. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
6;
9,  14;
12, 19, 26;
15, 24, 33, 42;
18, 29, 40, 51, 62;
21, 34, 47, 60, 73,  86;
24, 39, 54, 69, 84,  99,  114;
27, 44, 61, 78, 95,  112, 129, 146;
30, 49, 68, 87, 106, 125, 144, 163, 182;
33, 54, 75, 96, 117, 138, 159, 180, 201, 222; etc.

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

Cf. A153043, A020725, A016897, A017041.

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

t[n_,k_]:=2 n*k + n + k + 2; Table[t[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A182719 Numbers of the form 5k + 4, 7k + 5, or 11k + 6.

Original entry on oeis.org

4, 5, 6, 9, 12, 14, 17, 19, 24, 26, 28, 29, 33, 34, 39, 40, 44, 47, 49, 50, 54, 59, 61, 64, 68, 69, 72, 74, 75, 79, 82, 83, 84, 89, 94, 96, 99, 103, 104, 105, 109, 110, 114, 116, 117, 119, 124, 127, 129, 131, 134, 138, 139, 144, 145, 149, 152, 154, 159, 160, 164

Offset: 1

Omar E. Pol, Feb 08 2011

Numbers such that the Ramanujan congruences apply, making p(a(n)) divisible by at least one of 5, 7, or 11, where p is A000041.
Union of A016897, A017041 and A017461.
First differences are periodic with period length 145.

Wikipedia, Ramanujan's congruences
Index entries for linear recurrences with constant coefficients, order 146.

Cf. A000041, A016897, A017041, A017461.

IsA182719:=func< n | exists{ k: k in [0..n div 5] | n in [5*k+4, 7*k+5, 11*k+6] } >; [ n: n in [1..160] | IsA182719(n) ]; // Klaus Brockhaus, Feb 08 2011

Union[With[{no=30},Join[5Range[0,no]+4,7Range[0,no]+5,11Range[0,no]+6]]]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = a(n-145) + 385 = a(n-1) + a(n-145) - a(n-146).

Rewritten by Charles R Greathouse IV and Klaus Brockhaus, Feb 08 2011

A220502 spt(7n+5) where spt(n) = A092269(n).

Original entry on oeis.org

14, 238, 1820, 10486, 48692, 196168, 706671, 2335760, 7185780, 20832168, 57385734, 151261320, 383516700, 939524019, 2231714982, 5155845968, 11614187984, 25565740130, 55096236664, 116437751108, 241655216355, 493152387294, 990688365380

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

a(n) is divisible by 7 (see A220507).

G. E. Andrews, The number of smallest parts in the partitions of n

Cf. A017041, A092269, A213261, A220485, A220503, A220507.

a(n) = A092269(A017041(n)).

A220507 a(n) = spt(7n+5)/7 where spt(n) = A092269(n).

Original entry on oeis.org

2, 34, 260, 1498, 6956, 28024, 100953, 333680, 1026540, 2976024, 8197962, 21608760, 54788100, 134217717, 318816426, 736549424, 1659169712, 3652248590, 7870890952, 16633964444, 34522173765, 70450341042, 141526909340, 280158178412

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(7n+5) == 0 (mod 7) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220513.

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function

Cf. A017041, A071746, A092269, A220502, A220505, A220513.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[7 n + 5]/7;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A017041(n))/7 = A220502(n)/7.

cp's OEIS Frontend

A269100 a(n) = 13*n + 11.

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A125199 Triangle read by rows: T(n,k) = 4*n*k - n - k, 1<=k<=n.

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A047382 Numbers that are congruent to {0, 5} mod 7.

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A126980 a(n) = 14*n + 47.

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A155546 Triangle read by rows where T(m,n)=2mn + m + n - 5, 1<=n<=m.

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A163674 Triangle T(n,m) = 2*m*n + m + n + 9 read by rows.

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A155705 Triangle read by rows where T(m,n) = 2*m*n + m + n + 2.

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A125199 Triangle read by rows: T(n,k) = 4nk - n - k, 1<=k<=n.

A163674 Triangle T(n,m) = 2mn + m + n + 9 read by rows.

A155705 Triangle read by rows where T(m,n) = 2mn + m + n + 2.