%I #135 Nov 24 2022 05:22:04

%S 5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,

%T 131,137,143,149,155,161,167,173,179,185,191,197,203,209,215,221,227,

%U 233,239,245,251,257,263,269,275,281,287,293,299,305,311,317,323,329,335

%N a(n) = 6*n + 5.

%C Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(18).

%C Exponents e such that x^e + x - 1 is reducible.

%C First differences of A141631. - _Paul Curtz_, Sep 12 2008

%C a(n-1), n >= 1, appears as first column in the triangle A239127 related to the Collatz problem. - _Wolfdieter Lang_, Mar 14 2014

%C Odd unlucky numbers in A050505. - _Fred Daniel Kline_, Feb 25 2017

%C Intersection of A005408 and A016789. - _Bruno Berselli_, Apr 26 2018

%C Numbers that are not divisible by their digital root in base 4. - _Amiram Eldar_, Nov 24 2022

%H Muniru A Asiru, <a href="/A016969/b016969.txt">Table of n, a(n) for n = 0..3000</a>

%H Mark W. Coffey, <a href="http://arxiv.org/abs/1601.01673">Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers</a>, arXiv:1601.01673 [math.NT], 2016.

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

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=949">Encyclopedia of Combinatorial Structures 949</a>.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., Vol. 36, No. 3 (1935), pp. 637-649.

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4641886">The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1</a>, Politecnico di Torino (Italy, 2021).

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4662489">Binary operations inspired by generalized entropies applied to figurate numbers</a>, Politecnico di Torino (Italy, 2021).

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.

%H Leo Tavares, <a href="/A016969/a016969.jpg">Illustration: Twin Triangular Frames</a>.

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

%F a(n) = A003415(A003415(A125200(n+1)))/2. - _Reinhard Zumkeller_, Nov 24 2006

%F A008615(a(n)) = n+1. - _Reinhard Zumkeller_, Feb 27 2008

%F a(n) = A007310(2*n+1); complement of A016921 with respect to A007310. - _Reinhard Zumkeller_, Oct 02 2008

%F From _Klaus Brockhaus_, Jan 04 2009: (Start)

%F G.f.: (5+x)/(1-x)^2.

%F a(0) = 5; for n > 0, a(n) = a(n-1)+6.

%F (End)

%F a(n) = A016921(n)+4 = A016933(n)+3 = A016945(n)+2 = A016957(n)+1. - _Klaus Brockhaus_, Jan 04 2009

%F a(n) = floor((12n-1)/2) with offset 1..a(1)=5. - _Gary Detlefs_, Mar 07 2010

%F a(n) = 4*(3*n+1) - a(n-1) (with a(0) = 5). - _Vincenzo Librandi_, Nov 20 2010

%F a(n) = floor(1/(1/sin(1/n) - n)). - _Clark Kimberling_, Feb 19 2010

%F a(n) = 3*Sum_{k = 0..n} binomial(6*n+5, 6*k+2)*Bernoulli(6*k+2). - _Michel Marcus_, Jan 11 2016

%F a(n) = A049452(n+1) / (n+1). - _Torlach Rush_, Nov 23 2018

%F a(n) = 2*A000217(n+2) - 1 - 2*A000217(n-1). See Twin Triangular Frames illustration. - _Leo Tavares_, Aug 25 2021

%F Sum_{n>=0} (-1)^n/a(n) = Pi/6 - sqrt(3)*arccoth(sqrt(3))/3. - _Amiram Eldar_, Dec 10 2021

%t 6Range[0, 59] + 5 (* or *) NestList[6 + # &, 5, 60] (* _Harvey P. Dale_, Mar 09 2013 *)

%o (Magma) [ 6*n+5: n in [0..55] ]; // _Klaus Brockhaus_, Jan 04 2009

%o (PARI) a(n)=6*n+5 \\ _Charles R Greathouse IV_, Jul 10 2016

%o (Scala) (1 to 60).map(6 * _ - 1).mkString(", ") // _Alonso del Arte_, Nov 23 2018

%o (GAP) List([0..60],n->6*n+5); # _Muniru A Asiru_, Nov 24 2018

%Y Cf. A111863, A007310, A008588, A016921, A016933, A016945, A016957, A049452.

%Y Cf. A050505 (unlucky numbers).

%Y Cf. A000217.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _Klaus Brockhaus_, Jan 04 2009