A017137 -id:A017137 - cp's OEIS frontend

A004767 a(n) = 4*n + 3.

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 187, 191, 195, 199, 203, 207, 211, 215, 219, 223

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(12).
Binary expansion ends 11.
These the numbers for which zeta(2*x+1) needs just 2 terms to be evaluated. - Jorge Coveiro, Dec 16 2004 [This comment needs clarification]
a(n) is the smallest k such that for every r from 0 to 2n - 1 there exist j and i, k >= j > i > 2n - 1, such that j - i == r (mod (2n - 1)), with (k, (2n - 1)) = (j,(2n - 1)) = (i, (2n - 1)) = 1. - Amarnath Murthy, Sep 24 2003
Complement of A004773. - Reinhard Zumkeller, Aug 29 2005
Any (4n+3)-dimensional manifold endowed with a mixed 3-Sasakian structure is an Einstein space with Einstein constant lambda = 4n + 2 [Theorem 3, p. 10 of Ianus et al.]. - Jonathan Vos Post, Nov 24 2008
Solutions to the equation x^(2*x) = 3*x (mod 4*x). - Farideh Firoozbakht, May 02 2010
Subsequence of A022544. - Vincenzo Librandi, Nov 20 2010
First differences of A084849. - Reinhard Zumkeller, Apr 02 2011
Numbers n such that {1, 2, 3, ..., n} is a losing position in the game of Nim. - Franklin T. Adams-Watters, Jul 16 2011
Numbers n such that there are no primes p that satisfy the relationship p XOR n = p + n. - Brad Clardy, Jul 22 2012
The XOR of all numbers from 1 to a(n) is 0. - David W. Wilson, Apr 21 2013
A089911(4*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
First differences of A014105. - Ivan N. Ianakiev, Sep 21 2013
All triangular numbers in the sequence are congruent to {3, 7} mod 8. - Ivan N. Ianakiev, Nov 12 2013
Apart from the initial term, length of minimal path on an n-dimensional cubic lattice (n > 1) of side length 2, until a self-avoiding walk gets stuck. Construct a path connecting all 2n points orthogonally adjacent from the center, ending at the center. Starting at any point adjacent to the center, there are 2 steps to reach each of the remaining 2n - 1 points, resulting in path length 4n - 2 with a final step connecting the center, for a total path length of 4n - 1, comprising 4n points. - Matthew Lehman, Dec 10 2013
a(n-1), n >= 1, appears as first column in the triangles A238476 and A239126 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
For the Collatz Conjecture, we identify two types of odd numbers. This sequence contains all the ascenders: where (3*a(n) + 1) / 2 is odd and greater than a(n). See A016813 for the descenders. - Jaroslav Krizek, Jul 29 2016

			G.f. = 3 + 7*x + 11*x^2 + 15*x^3 + 19*x^4 + 23*x^5 + 27*x^6 + 31*x^7 + ...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 85.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999. See Theorem 8.1 on page 240.

Ivan Panchenko, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Stere Ianus, Mihai Visinescu and Gabriel-Eduard Vilcu, Hidden symmetries and Killing tensors on curved spaces, arXiv:0811.3478 [math-ph], 2008. - _Jonathan Vos Post_, Nov 24 2008
Tanya Khovanova, Recursive Sequences
Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, Generating Trees and Fibonacci Polyominoes, arXiv:2411.17812 [math.CO], 2024. See p. 8.
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. A004773, A005408, A008545 (partial products), A008586, A014105, A016813, A016825, A017137, A017629, A022544, A084849, A181049, A238476, A239126.
Cf. A017101 and A004771 (bisection: 3 and 7 mod 8).
Cf. A016838 (square).

a004767 = (+ 3) . (* 4)
a004767_list = [3, 7 ..]  -- Reinhard Zumkeller, Oct 03 2012

[4*n+3: n in [0..50]]; // Wesley Ivan Hurt, Jul 09 2014

Maple
```
seq( 3+4*n, n=0..100 );
```

4 Range[50] - 1 (* Wesley Ivan Hurt, Jul 09 2014 *)

a(n)=4*n+3 \\ Charles R Greathouse IV, Jul 28 2015

Vec((3+x)/(1-x)^2 + O(x^200)) \\ Altug Alkan, Jan 15 2016

for n in range(0,50): print(4*n+3, end=', ') # Stefano Spezia, Dec 12 2018

[4*n+3 for n in range(50)] # G. C. Greubel, Dec 09 2018

(0 to 59).map(4 *  + 3) // _Alonso del Arte, Dec 12 2018

G.f.: (3+x)/(1-x)^2. - Paul Barry, Feb 27 2003
a(n) = 2*a(n-1) - a(n-2) for n > 1, a(0) = 3, a(1) = 7. - Philippe Deléham, Nov 03 2008
a(n) = A017137(n)/2. - Reinhard Zumkeller, Jul 13 2010
a(n) = 8*n - a(n-1) + 2 for n > 0, a(0) = 3. - Vincenzo Librandi, Nov 20 2010
a(n) = A005408(A005408(n)). - Reinhard Zumkeller, Jun 27 2011
a(n) = 3 + A008586(n). - Omar E. Pol, Jul 27 2012
a(n) = A014105(n+1) - A014105(n). - Michel Marcus, Sep 21 2013
a(n) = A016813(n) + 2. - Jean-Bernard François, Sep 27 2013
a(n) = 4*n - 1, with offset 1. - Wesley Ivan Hurt, Mar 12 2014
From Ilya Gutkovskiy, Jul 29 2016: (Start)
E.g.f.: (3 + 4*x)*exp(x).
Sum_{n >= 0} (-1)^n/a(n) = (Pi + 2*log(sqrt(2) - 1))/(4*sqrt(2)) = A181049. (End)

A004771 a(n) = 8*n + 7. Or, numbers whose binary expansion ends in 111.

Original entry on oeis.org

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351, 359, 367, 375, 383, 391, 399, 407, 415, 423, 431

Offset: 0

N. J. A. Sloane

These numbers cannot be expressed as the sum of 3 squares. - Artur Jasinski, Nov 22 2006
These numbers cannot be perfect squares. - Cino Hilliard, Sep 03 2006
a(n-2), n >= 2, appears in the second column of triangle A239126 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
The initial terms 7, 15, 23, 31 are the generating set for the rest of the sequence in the sense that, by Lagrange's Four Square Theorem, any number n of the form 8*k+7 can always be written as a sum of no fewer than four squares, and if n = a^2 + b^2 + c^2 + d^2, then (a mod 4)^2 + (b mod 4)^2 + (c mod 4)^2 + (d mod 4)^2 must be one of 7, 15, 23, 31. - Walter Kehowski, Jul 07 2014
Define a set of consecutive positive odd numbers {1, 3, 5, ..., 12*n + 9} and skip the number 6*n + 5. Then the contraharmonic mean of that set gives this sequence. For example, ContraharmonicMean[{1, 3, 7, 9}] = 7. - Hilko Koning, Aug 27 2018
Jacobi symbol (2, a(n)) = Kronecker symbol (a(n), 2) = 1. - Jianing Song, Aug 28 2018

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 246.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Leo Tavares, Illustration: Twin Square Frames
Leo Tavares, Illustration: Mid-line Hexagons
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 962
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004767, A008590, A017077, A017101, A017137, A033954.
Cf. A007522 (primes), subsequence of A047522.
Cf. A056753, A227144, A227146, A239126.

List([0..60],n->8*n+7); # Muniru A Asiru, Aug 28 2018

a004771 = (+ 7) . (* 8)
a004771_list = [7, 15 ..]  -- Reinhard Zumkeller, Jan 29 2013

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

A004771:=n->8*n+7; seq(A004771(n), n=0..100); # Wesley Ivan Hurt, Dec 22 2013

8 Range[0, 60] + 7 (* or *) Range[7, 500, 8] (* or *) Table[8 n + 7, {n, 0, 60}] (* Bruno Berselli, Dec 28 2016 *)

a(n)=8*n+7 \\ Charles R Greathouse IV, Sep 23 2012

O.g.f: (7 + x)/(1 - x)^2 = 8/(1 - x)^2 - 1/(1 - x). - R. J. Mathar, Nov 30 2007
a(n) = 2*a(n-1) - a(n-2) for n >= 2.  - Vincenzo Librandi, May 28 2011
A056753(a(n)) = 7. - Reinhard Zumkeller, Aug 23 2009
a(n) = t(t(t(n))), where t(i) = 2*i + 1.
a(n) = A004767(2*n+1), for n >= 0. See also A004767(2*n) = A017101(n). - Wolfdieter Lang, Feb 03 2022
From Elmo R. Oliveira, Apr 11 2024: (Start)
E.g.f.: exp(x)*(7 + 8*x).
a(n) = A033954(n+1) - A033954(n). (End)

A008579 Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane.

Original entry on oeis.org

1, 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, 54, 68, 62, 78, 70, 88, 78, 98, 86, 108, 94, 118, 102, 128, 110, 138, 118, 148, 126, 158, 134, 168, 142, 178, 150, 188, 158, 198, 166, 208, 174, 218, 182, 228, 190, 238, 198, 248, 206, 258, 214, 268, 222, 278

Offset: 0

N. J. A. Sloane

Interesting because coefficients never become monotonic.
Also the coordination sequence for a planar net made of densely packed circles. - Yuriy Sibirmovsky, Sep 11 2016
Described by J.-G. Eon (2014) as the coordination sequence of the Kagome net. - N. J. A. Sloane, Jan 03 2018

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 161 (but beware errors).

T. D. Noe, Table of n, a(n) for n = 0..1000
Pierre de La Harpe, and P. I. Grigorchuk, Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook, Algebra and Logic 37.6 (1998): 353-356.
Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See p. 51.
Jean-Guillaume Eon, Topological density of nets: a direct calculation, Acta Crystallographica Section A (Foundations of Crystallography), A60 (2014), 7-18; DOI: 10.1107/S0108767303022037. See Section 5.
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 4.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, kgm
Yuriy Sibirmovsky, Illustration of initial terms with densely packed circles.
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Cf. A017137, A017365.

a008579 0 = 1
a008579 1 = 4
a008579 n = (10 - 2*m) * n' + 8*m - 2 where (n',m) = divMod n 2
a008579_list = 1 : 4 : concatMap (\x -> map (* 2) [5*x-1,4*x+3]) [1..]
-- Reinhard Zumkeller, Nov 12 2012

f := n->if n mod 2 = 0 then 10*(n/2)-2 else 8*(n-1)/2+6 fi;

a[n_?EvenQ] := 10*n/2-2; a[n_?OddQ] := 8*(n-1)/2+6; a[0] = 1; a[1] = 4; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Nov 18 2011, after Maple *)
CoefficientList[Series[(1+2x)(1+2x+2x^2+2x^3-x^4)/(1-x^2)^2,{x,0,50}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,4,8,14,18,22},50] (* Harvey P. Dale, Sep 05 2018 *)

G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4)/(1 - x^2)^2.
From R. J. Mathar, Nov 26 2014: (Start)
a(2n) = A017365(n), n > 0.
a(2n+1) = A017137(n), n > 0. (End)
From Stefano Spezia, Aug 07 2022: (Start)
a(n) = (9 + (-1)^n)*n/2 - 2*(-1)^n for n > 1.
E.g.f.: 3 - 2*x + (4*x - 2)*cosh(x) + (5*x + 2)*sinh(x). (End)

A017245 a(n) = 9*n + 7.

Original entry on oeis.org

7, 16, 25, 34, 43, 52, 61, 70, 79, 88, 97, 106, 115, 124, 133, 142, 151, 160, 169, 178, 187, 196, 205, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 304, 313, 322, 331, 340, 349, 358, 367, 376, 385, 394, 403, 412, 421, 430, 439, 448, 457, 466, 475, 484

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 7. - Halfdan Skjerning, Mar 15 2018

Tanya Khovanova, Recursive Sequences
J. Laroche and N. J. A. Sloane, Correspondence, 1977
Leo Tavares, Illustration: Triple Triangular Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A017233.

[9*n+7: n in [0..60]]; // Vincenzo Librandi, Apr 30 2015

Range[7, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
Table[9 n + 7, {n, 0, 70}] (* or *) CoefficientList[Series[(7 + 2 x)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Apr 30 2015 *)
LinearRecurrence[{2,-1},{7,16},60] (* Harvey P. Dale, Jul 30 2024 *)

vector(100,n,9*n-2) \\ Derek Orr, Apr 30 2015

a(n)^2 = A156676(n+1) + A017137(n). - Reinhard Zumkeller, Jul 13 2010
From Vincenzo Librandi, Apr 30 2015: (Start)
G.f.: (7+2*x)/(1-x)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
E.g.f.: exp(x)*(7 + 9*x). - Stefano Spezia, Dec 08 2024

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

Original entry on oeis.org

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

A089911 a(n) = Fibonacci(n) mod 12.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1

Offset: 0

Casey Mongoven, Nov 14 2003

From Reinhard Zumkeller, Jul 05 2013: (Start)
Sequence has been applied by several composers to 12-tone equal temperament pitch structure. The complete Fibonacci mod 12 system (a set of 10 periodic sequences) exhausts all possible ordered dyads; that is, every possible combination of two pitches is found in these sets.
a(A008594(n)) = 0;
a(A227144(n)) = 1;
a(3*A047522(n)) = 2;
a(A017569(n)) = a(2*A016933(n)) = a(4*A016777(n)) = 3;
a(2*A017629(n)) = a(3*A017137(n)) = a(6*A004767(n)) = 4;
a(A227146(n)) = 5;
a(nonexistent) = 6;
a(2*A017581(n)) = 7;
a(2*A017557(n)) = a(4*A016813(n)) = 8;
a(A017617(n)) = a(2*A016957(n)) = a(4*A016789(n)) = 9;
a(3*A047621(n)) = 10;
a(2*A017653(n)) = 11. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1199
Ron Knott, Fibonacci Numbers and the Golden Section
M. Renault, The Fibonacci Sequence Modulo M
A. P. Shah, Fibonacci Sequence Modulo m, Fibonacci Quarterly, Vol.6, No.2 (1968), 139-141.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1).

Cf. A000045, A001175, A003893, A010881.

a089911 n = a089911_list !! n
a089911_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 12)
                       (tail a089911_list) a089911_list
-- Reinhard Zumkeller, Jul 01 2013

[Fibonacci(n) mod 12: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

with(combinat,fibonacci); A089911 := proc(n) fibonacci(n) mod 12; end;

Table[Mod[Fibonacci[n], 12], {n, 0, 100}] (* Vincenzo Librandi, Feb 04 2014 *)

a(n)=fibonacci(n)%12 \\ Charles R Greathouse IV, Feb 03 2014

Has period of 24, restricted period 12 and multiplier 5.

a(n) = (a(n-1) + a(n-2)) mod 12, a(0) = 0, a(1) = 1.

More terms from Ray Chandler, Nov 15 2003

A047398 Numbers that are congruent to {3, 6} mod 8.

Original entry on oeis.org

3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, 46, 51, 54, 59, 62, 67, 70, 75, 78, 83, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 126, 131, 134, 139, 142, 147, 150, 155, 158, 163, 166, 171, 174, 179, 182, 187, 190, 195, 198, 203, 206, 211, 214, 219, 222, 227, 230

Offset: 1

N. J. A. Sloane

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

Union of A017101 and A017137.

Cf. A047461, A047452, A047470, A047524, A047535, A047615, A047617.

A047398:=n->4*n-(3+(-1)^n)/2: seq(A047398(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2017

Flatten[# + {3, 6} & /@ (8 Range[0, 28])] (* Arkadiusz Wesolowski, Sep 25 2012 *)
LinearRecurrence[{1,1,-1},{3,6,11},60] (* Harvey P. Dale, Oct 26 2020 *)

makelist(4*n + mod(n, 2) - 2, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */

def A047398(n): return ((n<<2)|(n&1))-2 # Chai Wah Wu, Mar 30 2024

a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Dec 05 2011: (Start)
a(n) = 4*n - (3 + (-1)^n)/2.
G.f.: x*(3+3*x+2*x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n > 3.
a(n) = 4*n + (n mod 2) - 2.
a(n) = A047470(n) + 3.
a(2*n) = A017137(n-1).
a(2*n-1) = A017101(n-1).
E.g.f.: ((8*x - 3)*exp(x) - exp(-x) + 4)/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 + log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A047451 Numbers that are congruent to {0, 6} mod 8.

Original entry on oeis.org

0, 6, 8, 14, 16, 22, 24, 30, 32, 38, 40, 46, 48, 54, 56, 62, 64, 70, 72, 78, 80, 86, 88, 94, 96, 102, 104, 110, 112, 118, 120, 126, 128, 134, 136, 142, 144, 150, 152, 158, 160, 166, 168, 174, 176, 182, 184, 190, 192, 198, 200, 206, 208, 214, 216, 222, 224, 230

Offset: 1

N. J. A. Sloane

All even numbers m such that Integral_{x=0..2*Pi} Product_{i=1..m/2} cos(2*i*x) dx is nonzero. - William Boyles, Oct 12 2019

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

Union of A008590 and A017137.

Cf. A030308, A047504 (complement).

Array[8 # + {0, 6} &, 29, 0] // Flatten (* or *)
Rest@ CoefficientList[Series[2 x^2*(3 + x)/((1 + x) (x - 1)^2), {x, 0, 58}], x] (* Michael De Vlieger, Nov 18 2019 *)
LinearRecurrence[{1,1,-1},{0,6,8},80] (* Harvey P. Dale, Apr 09 2022 *)

forstep(n=0,200,[6,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = 4*n - 3 + (-1)^n; \\ David Lovler, Jul 25 2022

a(n) = 8*n - a(n-1) - 10 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 3 + (-1)^n.
G.f.: 2*x^2*(3+x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011
a(n) = ceiling((8/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - Pi/16. - Amiram Eldar, Dec 18 2021
E.g.f.: (4*x + 1)*exp(x) - exp(-x) = 4*x*exp(x) + 2*sinh(x). - David Lovler, Aug 02 2022

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

Original entry on oeis.org

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

A047406 Numbers that are congruent to {4, 6} mod 8.

Original entry on oeis.org

4, 6, 12, 14, 20, 22, 28, 30, 36, 38, 44, 46, 52, 54, 60, 62, 68, 70, 76, 78, 84, 86, 92, 94, 100, 102, 108, 110, 116, 118, 124, 126, 132, 134, 140, 142, 148, 150, 156, 158, 164, 166, 172, 174, 180, 182, 188, 190, 196, 198, 204, 206, 212, 214, 220, 222, 228

Offset: 1

N. J. A. Sloane

In groups of four, add the odd and even numbers (4=1+3, 6=2+4; 12=5+7, 14=6+8; etc.). - George E. Antoniou, Dec 12 2001

The first 250 terms (4 through 998) are the 250 non-occurring Fibonacci number residues modulo 1000; i.e., if leading zeros are supplied as necessary for the terms having fewer than three digits, these are the 250 sets of three digits that never appear as the last three digits of a Fibonacci number. - Jon E. Schoenfield, Jul 05 2010

			a(2) = 8*2 - 4 - 6 = 6;
a(3) = 8*3 - 6 - 6 = 12;
a(4) = 8*4 - 12 - 6 = 14.

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

Union of A017113 and A017137.

Cf. A042964.

Select[Range[230], MemberQ[{4, 6}, Mod[#, 8]] &] (* Amiram Eldar, Dec 19 2021 *)

a(n)=4*n-1-(-1)^n \\ Charles R Greathouse IV, May 20 2013

a(n) = A042964(n)*2.
a(n) = (4*n - 1 - (-1)^n). - Jon E. Schoenfield, Jul 05 2010
a(n) = 8*n - a(n-1) - 6 (with a(1)=4). - Vincenzo Librandi, Aug 05 2010
G.f.: 2*x*(2+x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 18 2013: (Start)
a(n) = (8 * ceiling(n/2) - 4) * (n mod 2) + (8 * ceiling(n/2) - 2) * (n+1 mod 2).
a(n) = 8 * ceiling(n/2) - 3 + (-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - log(2)/8. - Amiram Eldar, Dec 19 2021
E.g.f.: 2*(1 + 2*x*exp(x) - cosh(x)). - David Lovler, Sep 02 2022

cp's OEIS Frontend

A004767 a(n) = 4*n + 3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Sage

Scala

Formula

A004771 a(n) = 8*n + 7. Or, numbers whose binary expansion ends in 111.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Formula

A008579 Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A017245 a(n) = 9*n + 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Sage

Formula

A089911 a(n) = Fibonacci(n) mod 12.

Views

Author

Keywords

Comments