



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
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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


REFERENCES

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 kcommuting permutations, arXiv preprint arXiv:1406.3081, 2014
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Wikipedia, Interval arithmetic
Index to sequences with linear recurrences with constant coefficients, signature (2,1).


FORMULA

a(n) = A005408(2*n).
sum(n=1, inf, (1)^n/a(n)) = 1/4/sqrt(2)*(Pi+2ln(sqrt(2)+1)) = A181048 [Jolley].  Benoit Cloitre, Apr 05 2002
G.f.: (1+3*x)/(1x)^2  Paul Barry, Feb 27 2003
(1 + 5x + 9x^2 + 13x^3...) = (1 + 2x + 3x^2...) / (1  3x + 9x^2 27x^3...)  Gary W. Adamson, Jul 03 2003
a(n) = A001969(n) + A000069(n) .  Philippe Deléham, Feb 04 2004
a(n) = A004766(n1). [R. J. Mathar, Oct 26 2008]
a(n) = 2*a(n1)a(n2); a(0)=1, a(1)=5. a(n)=4+a(n1). [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*n2a(n1) (with a(0)=1).  Vincenzo Librandi, Nov 20 2010
a(n) = A004767(n)2.  JeanBernard François, Sep 27 2013


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


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



