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 A016813 a(n) = 4*n + 1. 219
 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 k such that k and (k+1) have the same binary digital sum. - Benoit Cloitre, Jun 05 2002 Numbers k such that (1 + sqrt(k))/2 is an algebraic integer. - Alonso del Arte, Jun 04 2012 Numbers k such that 2 is the only prime p that satisfies the relationship p XOR k = p + k. - Brad Clardy, Jul 22 2012 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 A004767 for the ascenders. - Fred Daniel Kline, Nov 29 2014 [corrected by Jaroslav Krizek, Jul 29 2016] 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 For n > 3, also the number of (not necessarily maximum) cliques in the n-sunlet graph. - Eric W. Weisstein, Nov 29 2017 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 Starting with a(1) = 5, numbers ending with 01 in base 2. - John Keith, May 09 2022 REFERENCES K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 150. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Colin Defant and Noah Kravitz, Loops and Regions in Hitomezashi Patterns, arXiv:2201.03461 [math.CO], 2022. Theorem 1.3. L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16. 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") Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269. 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) Eric Weisstein's World of Mathematics, Clique Eric Weisstein's World of Mathematics, Hilbert Number Eric Weisstein's World of Mathematics, Sunlet Graph 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+2*log(sqrt(2)+1)) = A181048 [Jolley]. - Benoit Cloitre, Apr 05 2002 G.f.: (1+3*x)/(1-x)^2. - Paul Barry, Feb 27 2003 [corrected for offset 0 by Wolfdieter Lang, Oct 03 2014] (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 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 a(n) = A004767(n) - 2. - Jean-Bernard François, Sep 27 2013 a(n) = A058281(3n+1). - Eli Jaffe, Jun 07 2016 From Ilya Gutkovskiy, Jul 29 2016: (Start) E.g.f.: (1 + 4*x)*exp(x). a(n) = Sum_{k = 0..n} A123932(k). a(A005098(k)) = x^2 + y^2. Inverse binomial transform of A014480. (End) Dirichlet g.f.: 4*Zeta(-1 + s) + Zeta(s). - Stefano Spezia, Nov 02 2018 EXAMPLE From Leo Tavares, Jul 02 2021: (Start) Illustration of initial terms:                                         o                         o               o             o           o               o     o     o o o     o o o o o     o o o o o o o             o           o               o                         o               o                                         o (End) MAPLE seq(4*k+1, k=0..100); # Wesley Ivan Hurt, Sep 28 2013 MATHEMATICA Range[1, 237, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *) Table[4 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *) 4 Range[0, 20] + 1 (* Eric W. Weisstein, Nov 29 2017 *) LinearRecurrence[{2, -1}, {5, 9}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *) CoefficientList[Series[(1 + 3 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *) 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 (Scala) (0 to 59).map(4 * _ + 1) // Alonso del Arte, Aug 08 2018 (GAP) List([0..70], n->4*n+1); # Muniru A Asiru, Aug 08 2018 CROSSREFS a(n) = A093561(n+1, 1), (4, 1)-Pascal column. Cf. A016921, A017281, A017533, A047202, A158057, A161705, A161709, A161714, A128470. - Reinhard Zumkeller, Jun 17 2009 Subsequence of A042963. Cf. A004772 (complement). Cf. A017557. Sequence in context: A194395 A162502 A004766 * A314668 A314669 A334526 Adjacent sequences:  A016810 A016811 A016812 * A016814 A016815 A016816 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified September 26 08:00 EDT 2022. Contains 356987 sequences. (Running on oeis4.)