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A004767 a(n) = 4*n + 3.

%I A004767 #220 Jul 22 2025 15:50:28
%S A004767 3,7,11,15,19,23,27,31,35,39,43,47,51,55,59,63,67,71,75,79,83,87,91,
%T A004767 95,99,103,107,111,115,119,123,127,131,135,139,143,147,151,155,159,
%U A004767 163,167,171,175,179,183,187,191,195,199,203,207,211,215,219,223
%N A004767 a(n) = 4*n + 3.
%C A004767 Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(12).
%C A004767 Binary expansion ends 11.
%C A004767 These the numbers for which zeta(2*x+1) needs just 2 terms to be evaluated. - _Jorge Coveiro_, Dec 16 2004 [This comment needs clarification]
%C A004767 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
%C A004767 Complement of A004773. - _Reinhard Zumkeller_, Aug 29 2005
%C A004767 Any (4n+3)-dimensional manifold endowed with a mixed 3-Sasakian 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
%C A004767 Solutions to the equation x^(2*x) = 3*x (mod 4*x). - _Farideh Firoozbakht_, May 02 2010
%C A004767 Subsequence of A022544. - _Vincenzo Librandi_, Nov 20 2010
%C A004767 First differences of A084849. - _Reinhard Zumkeller_, Apr 02 2011
%C A004767 Numbers n such that {1, 2, 3, ..., n} is a losing position in the game of Nim. - _Franklin T. Adams-Watters_, Jul 16 2011
%C A004767 Numbers n such that there are no primes p that satisfy the relationship p XOR n = p + n. - _Brad Clardy_, Jul 22 2012
%C A004767 The XOR of all numbers from 1 to a(n) is 0. - _David W. Wilson_, Apr 21 2013
%C A004767 A089911(4*a(n)) = 4. - _Reinhard Zumkeller_, Jul 05 2013
%C A004767 First differences of A014105. - _Ivan N. Ianakiev_, Sep 21 2013
%C A004767 All triangular numbers in the sequence are congruent to {3, 7} mod 8. - _Ivan N. Ianakiev_, Nov 12 2013
%C A004767 Apart from the initial term, length of minimal path on an n-dimensional cubic lattice (n > 1) of side length 2, until a self-avoiding 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
%C A004767 a(n-1), n >= 1, appears as first column in the triangles A238476 and A239126 related to the Collatz problem. - _Wolfdieter Lang_, Mar 14 2014
%C A004767 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
%D A004767 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 85.
%D A004767 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999. See Theorem 8.1 on page 240.
%H A004767 Ivan Panchenko, <a href="/A004767/b004767.txt">Table of n, a(n) for n = 0..200</a>
%H A004767 Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H A004767 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H A004767 Stere Ianus, Mihai Visinescu and Gabriel-Eduard Vilcu, <a href="http://arxiv.org/abs/0811.3478">Hidden symmetries and Killing tensors on curved spaces</a>, arXiv:0811.3478 [math-ph], 2008. - _Jonathan Vos Post_, Nov 24 2008
%H A004767 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A004767 Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, <a href="https://arxiv.org/abs/2411.17812">Generating Trees and Fibonacci Polyominoes</a>, arXiv:2411.17812 [math.CO], 2024. See p. 8.
%H A004767 William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A004767 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H A004767 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A004767 G.f.: (3+x)/(1-x)^2. - _Paul Barry_, Feb 27 2003
%F A004767 a(n) = 2*a(n-1) - a(n-2) for n > 1, a(0) = 3, a(1) = 7. - _Philippe Deléham_, Nov 03 2008
%F A004767 a(n) = A017137(n)/2. - _Reinhard Zumkeller_, Jul 13 2010
%F A004767 a(n) = 8*n - a(n-1) + 2 for n > 0, a(0) = 3. - _Vincenzo Librandi_, Nov 20 2010
%F A004767 a(n) = A005408(A005408(n)). - _Reinhard Zumkeller_, Jun 27 2011
%F A004767 a(n) = 3 + A008586(n). - _Omar E. Pol_, Jul 27 2012
%F A004767 a(n) = A014105(n+1) - A014105(n). - _Michel Marcus_, Sep 21 2013
%F A004767 a(n) = A016813(n) + 2. - _Jean-Bernard François_, Sep 27 2013
%F A004767 a(n) = 4*n - 1, with offset 1. - _Wesley Ivan Hurt_, Mar 12 2014
%F A004767 From _Ilya Gutkovskiy_, Jul 29 2016: (Start)
%F A004767 E.g.f.: (3 + 4*x)*exp(x).
%F A004767 Sum_{n >= 0} (-1)^n/a(n) = (Pi + 2*log(sqrt(2) - 1))/(4*sqrt(2)) = A181049. (End)
%e A004767 G.f. = 3 + 7*x + 11*x^2 + 15*x^3 + 19*x^4 + 23*x^5 + 27*x^6 + 31*x^7 + ...
%p A004767 seq( 3+4*n, n=0..100 );
%t A004767 4 Range[50] - 1 (* _Wesley Ivan Hurt_, Jul 09 2014 *)
%o A004767 (Haskell)
%o A004767 a004767 = (+ 3) . (* 4)
%o A004767 a004767_list = [3, 7 ..]  -- _Reinhard Zumkeller_, Oct 03 2012
%o A004767 (Magma) [4*n+3: n in [0..50]]; // _Wesley Ivan Hurt_, Jul 09 2014
%o A004767 (PARI) a(n)=4*n+3 \\ _Charles R Greathouse IV_, Jul 28 2015
%o A004767 (PARI) Vec((3+x)/(1-x)^2 + O(x^200)) \\ _Altug Alkan_, Jan 15 2016
%o A004767 (Scala) (0 to 59).map(4 * _ + 3) // _Alonso del Arte_, Dec 12 2018
%o A004767 (Sage) [4*n+3 for n in range(50)] # _G. C. Greubel_, Dec 09 2018
%o A004767 (Python) for n in range(0,50): print(4*n+3, end=', ') # _Stefano Spezia_, Dec 12 2018
%Y A004767 Cf. A004773, A005408, A008545 (partial products), A008586, A014105, A016813, A016825, A017137, A017629, A022544, A084849, A181049, A238476, A239126.
%Y A004767 Cf. A017101 and A004771 (bisection: 3 and 7 mod 8).
%Y A004767 Cf. A016838 (square).
%K A004767 nonn,easy
%O A004767 0,1
%A A004767 _N. J. A. Sloane_