

A004767


a(n) = 4*n + 3.


139



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


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 (jorgecoveiro(AT)yahoo.com), 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 3Sasakian 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. AdamsWatters, 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 a ndimensional cubic lattice (n > 1) of side length 2, until a selfavoiding 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(n1), 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


LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..200
GuoNiu Han, Enumeration of Standard Puzzles
GuoNiu Han, Enumeration of Standard Puzzles [Cached copy]
Stere Ianus, Mihai Visinescu and GabrielEduard Vilcu, Hidden symmetries and Killing tensors on curved spaces, arXiv:0811.3478 [mathph], 2008.  Jonathan Vos Post, Nov 24 2008
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).


FORMULA

G.f.: (3+x)/(1x)^2.  Paul Barry, Feb 27 2003
a(n) = 2*a(n1)  a(n2) 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(n1) + 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.  JeanBernard 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)


EXAMPLE

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


MAPLE

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


MATHEMATICA

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


PROG

(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
(PARI) a(n)=4*n+3 \\ Charles R Greathouse IV, Jul 28 2015
(PARI) Vec((3+x)/(1x)^2 + O(x^200)) \\ Altug Alkan, Jan 15 2016
(Scala) (0 to 59).map(4 * _ + 3) // Alonso del Arte, Dec 12 2018
(Sage) [4*n+3 for n in range(50)] # G. C. Greubel, Dec 09 2018
(Python) for n in range(0, 50): print(4*n+3, end=', ') # Stefano Spezia, Dec 12 2018


CROSSREFS

Cf. A008586, A016813, A016825, A017629, A008545 (partial products).
Sequence in context: A103543 A172338 A189787 * A131098 A118894 A194397
Adjacent sequences: A004764 A004765 A004766 * A004768 A004769 A004770


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



