A089911 -id:A089911 - cp's OEIS frontend

A017629 a(n) = 12*n + 9.

9, 21, 33, 45, 57, 69, 81, 93, 105, 117, 129, 141, 153, 165, 177, 189, 201, 213, 225, 237, 249, 261, 273, 285, 297, 309, 321, 333, 345, 357, 369, 381, 393, 405, 417, 429, 441, 453, 465, 477, 489, 501, 513, 525, 537, 549, 561, 573, 585, 597, 609, 621, 633

Offset: 0

N. J. A. Sloane

Numbers k such that k mod 2 = (k+1) mod 3 = 1 and (k+2) mod 4 != 1. - Klaus Brockhaus, Jun 15 2004
For n > 3, the number of squares on the infinite 3-column chessboard at <= n knight moves from any fixed point. - Ralf Stephan, Sep 15 2004
A016946 is the subsequence of squares (for n = 3*k*(k+1) = A028896(k), then a(n) = (6k+3)^2 = A016946(k)). - Bernard Schott, Apr 05 2021

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

Subsequences include A072065, A016945, A016813, A008585, and A005408.

Cf. A008594, A017533, A017545, A017137, A004767, A001019, A028896, A016946, A089911.

a017629 = (+ 9) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

12*Range[0,200]+9 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
LinearRecurrence[{2,-1},{9,21},60] (* Harvey P. Dale, Apr 14 2019 *)

a(n)=12*n+9 \\ Charles R Greathouse IV, Jul 10 2016

[i+9 for i in range(525) if gcd(i,12) == 12] # Zerinvary Lajos, May 21 2009

a(n) = 6*(4*n+1) - a(n-1) (with a(0)=9). - Vincenzo Librandi, Dec 17 2010
A089911(2*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
G.f.: (9 + 3*x)/(1 - x)^2. - Alejandro J. Becerra Jr., Jul 08 2020
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(12*sqrt(2)). - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(3 + 4*x). - Stefano Spezia, Feb 25 2023

A047621 Numbers that are congruent to {3, 5} mod 8.

Original entry on oeis.org

3, 5, 11, 13, 19, 21, 27, 29, 35, 37, 43, 45, 51, 53, 59, 61, 67, 69, 75, 77, 83, 85, 91, 93, 99, 101, 107, 109, 115, 117, 123, 125, 131, 133, 139, 141, 147, 149, 155, 157, 163, 165, 171, 173, 179, 181, 187, 189, 195, 197, 203, 205, 211, 213, 219, 221, 227, 229

Offset: 1

N. J. A. Sloane

Numbers k for which Jacobi symbol J(2,k) = -1, so 2 (as well as 2^k) is not a square mod k. - Antti Karttunen, Aug 27 2005, corrected by Jianing Song, Nov 05 2019, see also A329095.

Numbers n whose multiplicative order modulo 2^k is 2^(k - 2) for k >= 4. For k = 3, the numbers whose multiplicative order modulo 8 is 2 are in sequence A047484. - Jianing Song, Apr 29 2018

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Row 1 of A112070. Complement of A047522 relative to A005408. Primes in this sequence: A003629.
Subsequence of A329095.
Cf. A047522, A066507, A101464, A144981.

a:=[3];; for n in [2..60] do a[n]:=8*n-a[n-1]-8; od; a; # Muniru A Asiru, Dec 04 2018

a047621 n = a047621_list !! (n-1)
a047621_list = 3 : 5 : map (+ 8) a047621_list
-- Reinhard Zumkeller, Jul 05 2013

LinearRecurrence[{1, 1, -1}, {3, 5, 11}, 100] (* Jean-François Alcover, Jul 31 2018 *)

a(n) = 8*n - a(n-1) - 8 (with a(1) = 3). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(3 + 2*x + 3*x^2) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Oct 08 2011
A089911(3*a(n)) = 10. - Reinhard Zumkeller, Jul 05 2013
a(n) = 8*floor((n - 1)/2) + 4 + (-1)^n. - Gary Detlefs, Dec 03 2018
From Franck Maminirina Ramaharo, Dec 03 2018: (Start)
a(n) = 4*n - 2 - (-1)^n.
E.g.f.: 3 - (2 - 4*x)*exp(x) - exp(-x). (End)
a(n + 2) = a(n) + 8. - David A. Corneth, Dec 03 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8. - Amiram Eldar, Dec 11 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/8) (1/A144981).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/8) (A101464). (End)

A017569 a(n) = 12*n + 4.

Original entry on oeis.org

4, 16, 28, 40, 52, 64, 76, 88, 100, 112, 124, 136, 148, 160, 172, 184, 196, 208, 220, 232, 244, 256, 268, 280, 292, 304, 316, 328, 340, 352, 364, 376, 388, 400, 412, 424, 436, 448, 460, 472, 484, 496, 508, 520, 532, 544, 556, 568, 580, 592, 604, 616, 628

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(46).
Number of 6 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1A008574; m=3: A016933; m=4: A022144; m=5: A017293. - Sergey Kitaev, Nov 13 2004
Except for 4, exponents e such that x^e - x^2 + 1 is reducible.
If Y and Z are 2-blocks of a (3n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
Terms are perfect squares iff n is a generalized octagonal number (A001082), then n = k*(3*k-2) and a(n) = (2*(3*k-1))^2. - Bernard Schott, Feb 26 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A017533, A017545, A089911.
Cf. A016777, A016933, A017293, A022144.
Cf. A001082.

a017569 = (+ 4) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+4: n in [0..50]]; // Vincenzo Librandi, May 04 2011

12*Range[0,200]+4 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

A089911(a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/12. - Amiram Eldar, Dec 12 2021
From Stefano Spezia, Feb 25 2023: (Start)
O.g.f.: 4*(1 + 2*x)/(1 - x)^2.
E.g.f.: 4*exp(x)*(1 + 3*x). (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = 2*A016933(n) = 4*A016777(n) = A016777(4*n+1). (End)

A017617 a(n) = 12*n + 8.

Original entry on oeis.org

8, 20, 32, 44, 56, 68, 80, 92, 104, 116, 128, 140, 152, 164, 176, 188, 200, 212, 224, 236, 248, 260, 272, 284, 296, 308, 320, 332, 344, 356, 368, 380, 392, 404, 416, 428, 440, 452, 464, 476, 488, 500, 512, 524, 536, 548, 560, 572, 584, 596, 608, 620, 632, 644, 656

Offset: 0

N. J. A. Sloane

Also the number of cube units that frame a cube of edge length n+1. - Peter M. Chema, Mar 27 2016

			For n=3; a(3)= 12*3 + 8 = 44.
Thus, there are 44 cube units that frame a cube of edge length 4. - _Peter M. Chema_, Mar 26 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A016789, A016933, A016957, A017533, A017545, A089911.

a017617 = (+ 8) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+8: n in [0..60]]; // Vincenzo Librandi, Jun 08 2011

12*Range[0,200]+8 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

my(x='x+O('x^99)); Vec(4*(2+x)/(1-x)^2) \\ Altug Alkan, Mar 27 2016

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 08 2011
A089911(a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
G.f.: 12*x/(1-x)^2 + 8/(1-x) = 4*(2+x)/(1-x)^2. (see the PARI program). - Wolfdieter Lang, Oct 11 2021
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/36 - log(2)/12. - Amiram Eldar, Dec 12 2021
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 4*exp(x)*(2 + 3*x).
a(n) = 4*A016789(n) = 2*A016957(n) = A016933(2*n+1). (End)

A017557 a(n) = 12*n + 3.

Original entry on oeis.org

3, 15, 27, 39, 51, 63, 75, 87, 99, 111, 123, 135, 147, 159, 171, 183, 195, 207, 219, 231, 243, 255, 267, 279, 291, 303, 315, 327, 339, 351, 363, 375, 387, 399, 411, 423, 435, 447, 459, 471, 483, 495, 507, 519, 531, 543, 555, 567, 579, 591, 603, 615, 627, 639

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 44 ).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A016813, A017533, A017545, A089911.

List([0..60], n-> 12*n+3 ); # G. C. Greubel, Sep 18 2019

a017557 = (+ 3) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+3: n in [0..60]]; // Vincenzo Librandi, Jun 07 2011

seq(12*n+3, n=0..60); # G. C. Greubel, Sep 18 2019

12*Range[0,60]+3 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=12*n+3 \\ Charles R Greathouse IV, Jul 10 2016

[12*n+3 for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 07 2011
A089911(2*a(n)) = 8. - Reinhard Zumkeller, Jul 05 2013
From G. C. Greubel, Sep 18 2019: (Start)
G.f.: 3*(1+3*x)/(1-x)^2.
E.g.f.: 3*(1+4*x)*exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = (Pi + 2*log(sqrt(2)+1))/(12*sqrt(2)). - Amiram Eldar, Dec 12 2021

A017653 a(n) = 12*n + 11.

Original entry on oeis.org

11, 23, 35, 47, 59, 71, 83, 95, 107, 119, 131, 143, 155, 167, 179, 191, 203, 215, 227, 239, 251, 263, 275, 287, 299, 311, 323, 335, 347, 359, 371, 383, 395, 407, 419, 431, 443, 455, 467, 479, 491, 503, 515, 527, 539, 551, 563, 575, 587, 599, 611, 623, 635

Offset: 0

N. J. A. Sloane

Or, with a different offset, 12*n - 1. In any case, numbers congruent to -1 (mod 12). - Alonso del Arte, May 29 2011

Numbers congruent to 2 (mod 3) and 3 (mod 4). - Bruno Berselli, Jul 06 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
John Elias, 60*n+55 Triangular Snowflakes.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1000.
Tanya Khovanova, Recursive Sequences.
Leo Tavares, Illustration: Twin Hexagonal Frames.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003215, A008594, A016969, A017533, A017545, A089911.

Subsequence of A072065.

a017653 = (+ 11) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+11: n in [0..60]]; // Vincenzo Librandi, Jun 08 2011

Array[12*#+11&,100,0] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)

PARI
```
a(n)=12*n+11
```

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 08 2011
G.f.: (11+x)/(1-x)^2. - Colin Barker, Feb 19 2012
A089911(2*a(n)) = 11. - Reinhard Zumkeller, Jul 05 2013
a(n) = 2*A003215(n+1) - 1 - 2*A003215(n). See Twin Hexagonal Frames illustration. - Leo Tavares, Aug 19 2021
From Elmo R. Oliveira, Apr 12 2025: (Start)
E.g.f.: exp(x)*(11 +  12*x).
a(n) = A016969(2*n+1). (End)

A017581 a(n) = 12*n + 5.

Original entry on oeis.org

5, 17, 29, 41, 53, 65, 77, 89, 101, 113, 125, 137, 149, 161, 173, 185, 197, 209, 221, 233, 245, 257, 269, 281, 293, 305, 317, 329, 341, 353, 365, 377, 389, 401, 413, 425, 437, 449, 461, 473, 485, 497, 509, 521, 533, 545, 557, 569, 581, 593, 605, 617, 629

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(71).
A089911(2*a(n)) = 7. - Reinhard Zumkeller, Jul 05 2013
Equivalently, intersection of A016813 and A016789. - Bruno Berselli, Jan 24 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A016789, A016813, A016969, A017533, A017545, A017593, A089911.

a017581 = (+ 5) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+5: n in [0..60]]; // Vincenzo Librandi, Jun 08 2011

12*Range[0,200]+5 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=12*n+5 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 2*a(n-1) - a(n-2) for n>1, a(0)=5, a(1)=17. - Vincenzo Librandi, Jun 08 2011
G.f.: x*(5 + 7*x)/(1 - x)^2. - Wolfdieter Lang, Jul 04 2023
E.g.f.: exp(x)*(5 + 12*x). - Stefano Spezia, Feb 21 2024
a(n) = A016969(2*n) = A016789(4*n+1). - Elmo R. Oliveira, Apr 10 2025

A227144 Numbers that are congruent to {1, 2, 7, 17, 23} modulo 24.

Original entry on oeis.org

1, 2, 7, 17, 23, 25, 26, 31, 41, 47, 49, 50, 55, 65, 71, 73, 74, 79, 89, 95, 97, 98, 103, 113, 119, 121, 122, 127, 137, 143, 145, 146, 151, 161, 167, 169, 170, 175, 185, 191, 193, 194, 199, 209, 215, 217, 218, 223, 233, 239, 241, 242, 247, 257, 263, 265, 266

Offset: 1

Reinhard Zumkeller, Jul 05 2013

A089911(a(n)) = 1.

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

Cf. A004771, A089911, A227146.

a227144 n = a227144_list !! (n-1)
a227144_list = [1,2,7,17,23] ++ map (+ 24) a227144_list

[n : n in [0..300] | n mod 24 in [1, 2, 7, 17, 23]]; // Wesley Ivan Hurt, Dec 26 2016

A227144:=n->24*floor(n/5)+[1, 2, 7, 17, 23][(n mod 5)+1]: seq(A227144(n), n=0..100); # Wesley Ivan Hurt, Dec 26 2016

Select[Range[500], MemberQ[{1, 2, 7, 17, 23}, Mod[#, 24]] &] (* Wesley Ivan Hurt, Dec 26 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{1,2,7,17,23,25},60] (* Harvey P. Dale, Dec 18 2019 *)

Vec(x*(1+x)*(x^4 +5*x^3 +5*x^2 +1)/((x^4 +x^3 +x^2 +x +1)*(x-1)^2) + O(x^50)) \\ G. C. Greubel, Dec 26 2016

G.f.: x*(1+x)*(x^4+5*x^3+5*x^2+1) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Jul 17 2013
From Wesley Ivan Hurt, Dec 26 2016: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
a(n) = (120*n - 110 - 6*(n mod 5) - 26*((n+1) mod 5) - ((n+2) mod 5) + 19*((n+3) mod 5) + 14*((n+4) mod 5))/25.
a(5k) = 24k-1, a(5k-1) = 24k-7, a(5k-2) = 24k-17, a(5k-3) = 24k-22, a(5k-4) = 24k-23. (End)

A227146 Numbers that are congruent to {5, 11, 13, 14, 19} modulo 24.

Original entry on oeis.org

5, 11, 13, 14, 19, 29, 35, 37, 38, 43, 53, 59, 61, 62, 67, 77, 83, 85, 86, 91, 101, 107, 109, 110, 115, 125, 131, 133, 134, 139, 149, 155, 157, 158, 163, 173, 179, 181, 182, 187, 197, 203, 205, 206, 211, 221, 227, 229, 230, 235, 245, 251, 253, 254, 259, 269

Offset: 1

Reinhard Zumkeller, Jul 05 2013

A089911(a(n)) = 5.

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

Cf. A004771, A089911, A227144.

a227146 n = a227146_list !! (n-1)
a227146_list = [5,11,13,14,19] ++ map (+ 24) a227146_list

Select[Range[300],MemberQ[{5,11,13,14,19},Mod[#,24]]&] (* or *) LinearRecurrence[{1,0,0,0,1,-1},{5,11,13,14,19,29},60] (* Harvey P. Dale, Apr 30 2018 *)

G.f.: x*(1+x)*(5*x^4+x^2+x+5) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Jul 17 2013
From Wesley Ivan Hurt, Dec 28 2016: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
a(n) = (120*n - 50 - (n mod 5) + 19*((n+1) mod 5) + 14*((n+2) mod 5) - 6*((n+3) mod 5) - 26*((n+4) mod 5))/25. (End)

cp's OEIS Frontend

A017629 a(n) = 12*n + 9.

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A047621 Numbers that are congruent to {3, 5} mod 8.

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A017569 a(n) = 12*n + 4.

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A017617 a(n) = 12*n + 8.

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A017557 a(n) = 12*n + 3.

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A017653 a(n) = 12*n + 11.

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