A016813 - OEIS

%I #283 Jul 16 2024 11:57:01

%S 1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,69,73,77,81,85,89,93,

%T 97,101,105,109,113,117,121,125,129,133,137,141,145,149,153,157,161,

%U 165,169,173,177,181,185,189,193,197,201,205,209,213,217,221,225,229,233,237

%N a(n) = 4*n + 1.

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

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

%C Numbers k such that k and (k+1) have the same binary digital sum. - _Benoit Cloitre_, Jun 05 2002

%C Numbers k such that (1 + sqrt(k))/2 is an algebraic integer. - _Alonso del Arte_, Jun 04 2012

%C Numbers k such that 2 is the only prime p that satisfies the relationship p XOR k = p + k. - _Brad Clardy_, Jul 22 2012

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

%C     1,   9,  21,  37,  57,  81, ...

%C     5,  17,  33,  53,  77, 105, ...

%C    13,  29,  49,  73, 101, 133, ...

%C    25,  45,  69,  97, 129, 165, ...

%C    41,  65,  93, 125, 161, 201, ...

%C    61,  89, 121, 157, 197, 241, ...

%C    ...

%C - _R. J. Mathar_, Jul 10 2013

%C 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

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

%C 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 A004767 for the ascenders. - _Fred Daniel Kline_, Nov 29 2014 [corrected by _Jaroslav Krizek_, Jul 29 2016]

%C 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

%C For n > 3, also the number of (not necessarily maximal) cliques in the n-sunlet graph. - _Eric W. Weisstein_, Nov 29 2017

%C For integers k with absolute value in A047202, also exponents of the powers of k having the same unit digit of k in base 10. - _Stefano Spezia_, Feb 23 2021

%C Starting with a(1) = 5, numbers ending with 01 in base 2. - _John Keith_, May 09 2022

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

%H Vincenzo Librandi, <a href="/A016813/b016813.txt">Table of n, a(n) for n = 0..1000</a>

%H Colin Defant and Noah Kravitz, <a href="https://arxiv.org/abs/2201.03461">Loops and Regions in Hitomezashi Patterns</a>, arXiv:2201.03461 [math.CO], 2022. Theorem 1.3.

%H L. B. W. Jolley, <a href="https://www.scribd.com/document/22916829/Summation-of-Series-2nd-rev-ed-L-B-W-Jolley-1961-WW">Summation of Series</a>, Dover Publications, 1961, p. 16.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Konrad Knopp, <a href="http://name.umdl.umich.edu/ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")

%H Konrad Knopp, <a href="https://archive.org/details/theoryandapplica031692mbp/mode/2up">Theory and Application of Infinite Series</a>, Dover, p. 269.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

%H William A. Stein's The Modular Forms Database, <a href="http://wstein.org/Tables/dimensions.html">PARI-readable dimension tables for Gamma_0(N)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertNumber.html">Hilbert Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Interval_arithmetic">Interval arithmetic</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

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

%F Sum_{n>=0} (-1)^n/a(n) = (1/(4*sqrt(2)))*(Pi+2*log(sqrt(2)+1)) = A181048 [Jolley]. - _Benoit Cloitre_, Apr 05 2002 [corrected by _Amiram Eldar_, Jul 30 2023]

%F G.f.: (1+3*x)/(1-x)^2. - _Paul Barry_, Feb 27 2003 [corrected for offset 0 by _Wolfdieter Lang_, Oct 03 2014]

%F (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

%F a(n) = A001969(n) + A000069(n). - _Philippe Deléham_, Feb 04 2004

%F a(n) = A004766(n-1). - _R. J. Mathar_, Oct 26 2008

%F 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

%F A056753(a(n)) = 3. - _Reinhard Zumkeller_, Aug 23 2009

%F A179821(a(n)) = a(A179821(n)). - _Reinhard Zumkeller_, Jul 31 2010

%F a(n) = 8*n - 2 - a(n-1) for n > 0, a(0) = 1. - _Vincenzo Librandi_, Nov 20 2010

%F 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

%F A089911(6*a(n)) = 8. - _Reinhard Zumkeller_, Jul 05 2013

%F a(n) = A004767(n) - 2. - _Jean-Bernard François_, Sep 27 2013

%F a(n) = A058281(3n+1). - _Eli Jaffe_, Jun 07 2016

%F From _Ilya Gutkovskiy_, Jul 29 2016: (Start)

%F E.g.f.: (1 + 4*x)*exp(x).

%F a(n) = Sum_{k = 0..n} A123932(k).

%F a(A005098(k)) = x^2 + y^2.

%F Inverse binomial transform of A014480. (End)

%F Dirichlet g.f.: 4*Zeta(-1 + s) + Zeta(s). - _Stefano Spezia_, Nov 02 2018

%e From _Leo Tavares_, Jul 02 2021: (Start)

%e Illustration of initial terms:

%e                                         o

%e                         o               o

%e             o           o               o

%e     o     o o o     o o o o o     o o o o o o o

%e             o           o               o

%e                         o               o

%e                                         o

%e (End)

%p seq(4*k+1, k=0..100); # _Wesley Ivan Hurt_, Sep 28 2013

%t Range[1, 237, 4] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)

%t Table[4 n + 1, {n, 0, 20}] (* _Eric W. Weisstein_, Nov 29 2017 *)

%t 4 Range[0, 20] + 1 (* _Eric W. Weisstein_, Nov 29 2017 *)

%t LinearRecurrence[{2, -1}, {5, 9}, {0, 20}] (* _Eric W. Weisstein_, Nov 29 2017 *)

%t CoefficientList[Series[(1 + 3 x)/(-1 + x)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)

%o (Magma) [n: n in [1..250 by 4]];

%o (Haskell)

%o a016813 = (+ 1) . (* 4)

%o a016813_list = [1, 5 ..]  -- _Reinhard Zumkeller_, Feb 14 2012

%o (PARI) a(n)=4*n+1 \\ _Charles R Greathouse IV_, Mar 22 2013

%o (PARI) x='x+O('x^100); Vec((1+3*x)/(1-x)^2) \\ _Altug Alkan_, Oct 22 2015

%o (Scala) (0 to 59).map(4 * _ + 1) // _Alonso del Arte_, Aug 08 2018

%o (GAP) List([0..70],n->4*n+1); # _Muniru A Asiru_, Aug 08 2018

%Y Subsequence of A042963 and of A079523.

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

%Y Cf. A016921, A017281, A017533, A047202, A158057, A161705, A161709, A161714, A128470. - _Reinhard Zumkeller_, Jun 17 2009

%Y Cf. A004772 (complement).

%Y Cf. A017557.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_