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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

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

References

• 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.

Links

• 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).

Crossrefs

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).

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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

PARI

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

PARI

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

Python

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

Sage

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

Scala

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

Formula

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)