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A016813 a(n) = 4n + 1. 162
1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229, 233, 237 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 23 ).

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 64 ).

Numbers n such that n and (n+1) have the same binary digital sum. - Benoit Cloitre, Jun 05 2002

Numbers n such that (1 + sqrt(n))/2 is an algebraic integer. - Alonso del Arte, Jun 04 2012

Numbers n such that 2 is the only prime p that satisfies the relationship p XOR n = p + n. - Brad Clardy, Jul 22 2012

The identity (4*n+1)^2-(4*n^2+2*n)*(2)^2 = 1 can be written as a(n)^2 - A002943(n)*2^2 = 1. - Vincenzo Librandi, Mar 11 2009 - Nov 25 2012

A089911(6*a(n)) = 8. - Reinhard Zumkeller, Jul 05 2013

This may also be interpreted as the array T(n,k) = A001844(n+k)+A008586(k) read by antidiagonals:

    1,   9,  21,  37,  57,  81, ...

    5,  17,  33,  53,  77, 105, ...

   13,  29,  49,  73, 101, 133, ...

   25,  45,  69,  97, 129, 165, ...

   41,  65,  93, 125, 161, 201, ...

   61,  89, 121, 157, 197, 241, ...

- R. J. Mathar, Jul 10 2013

With leading term 2 instead of 1, 1/a(n) is the largest tolerance of form 1/k, where k is a positive integer, so that the nearest integer to (n - 1/k)^2 and to (n + 1/k)^2 is n^2. In other words, if interval arithmetic is used to square [n - 1/k, n + 1/k], every value in the resulting interval of length 4n/k rounds to n^2 if and only if k >= a(n). - Rick L. Shepherd, Jan 20 2014

Odd numbers for which the number of prime factors congruent to 3 (mod 4) is even. - Daniel Forgues, Sep 20 2014

For the Collatz Conjecture, we identify two types of odd numbers. This sequence contains all the descenders: where (3*a(n) + 1) / 2 is even and requires additional divisions by 2. See A189787 for the ascenders. - Fred Daniel Kline, Nov 29 2014

a(n-1), n >=1, is also the complex dimension of the manifold M(S), the set of all conjugacy classes of irreducible representations of the fundamental group pi_1(X,x_0) of rank 2, where S = {a_1, ...,a_{n}, a_{n+1}=oo}, a subset of P^1 = C U {oo}, X = X(S) = P^1 \ S, and x_0 a base point in X. See the Iwasaki et al. reference, Proposition 2.1.4. p. 150.  - Wolfdieter Lang, Apr 22 2016

REFERENCES

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 150.

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 16.

Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

William A. Stein's The Modular Forms Database, PARI-readable dimension tables for Gamma_0(N)

Wikipedia, Interval arithmetic

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

FORMULA

a(n) = A005408(2*n).

Sum_{n>0} (-1)^n/a(n)) = 1/4/sqrt(2)*(Pi+2log(sqrt(2)+1)) = A181048 [Jolley]. - Benoit Cloitre, Apr 05 2002

G.f.: (1+3*x)/(1-x)^2. - Paul Barry, Feb 27 2003

(1 + 5*x + 9*x^2 + 13*x^3...) = (1 + 2*x + 3*x^2 + ...) / (1 - 3*x + 9*x^2 - 27*x^3+ ...) - Gary W. Adamson, Jul 03 2003

a(n) = A001969(n) + A000069(n). - Philippe Deléham, Feb 04 2004

a(n) = A004766(n-1). - R. J. Mathar, Oct 26 2008

a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=5. a(n) = 4 + a(n-1). - Philippe Deléham, Nov 03 2008

A056753(a(n)) = 3. - Reinhard Zumkeller, Aug 23 2009

A179821(a(n)) = a(A179821(n)). - Reinhard Zumkeller, Jul 31 2010

a(n) = 8*n - 2 - a(n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 20 2010

a(n) = A004767(n) - 2. - Jean-Bernard François, Sep 27 2013

a(n) = A058281(3n+1) - Eli Jaffe, Jun 07 2016

MAPLE

seq(4*k+1, k=0..100); # Wesley Ivan Hurt, Sep 28 2013

MATHEMATICA

Range[1, 500, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

PROG

(MAGMA) [n: n in [1..250 by 4]];

(Haskell)

a016813 = (+ 1) . (* 4)

a016813_list = [1, 5 ..]  -- Reinhard Zumkeller, Feb 14 2012

(PARI) a(n)=4*n+1 \\ Charles R Greathouse IV, Mar 22 2013

(PARI) x='x+O('x^100); Vec((1+3*x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015

CROSSREFS

a(n)= A093561(n+1, 1), (4, 1)-Pascal column.

Cf. A016921, A017281, A017533, A158057, A161705, A161709, A161714, A128470. [Reinhard Zumkeller, Jun 17 2009]

Subsequence of A042963.

Cf. A004772 (complement).

Cf. A017557.

Sequence in context: A194395 A162502 A004766 * A198395 A190951 A057948

Adjacent sequences:  A016810 A016811 A016812 * A016814 A016815 A016816

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Paul Barry's g.f. corrected for offset 0. - Wolfdieter Lang, Oct 03 2014

STATUS

approved

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Last modified July 25 00:31 EDT 2016. Contains 275024 sequences.