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%I #133 Sep 08 2022 08:44:41
%S 1,7,13,19,25,31,37,43,49,55,61,67,73,79,85,91,97,103,109,115,121,127,
%T 133,139,145,151,157,163,169,175,181,187,193,199,205,211,217,223,229,
%U 235,241,247,253,259,265,271,277,283,289,295,301,307,313,319,325,331
%N a(n) = 6*n + 1.
%C Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 22 ).
%C Also solutions to 2^x + 3^x == 5 (mod 7). - _Cino Hilliard_, May 10 2003
%C Except for 1, exponents n > 1 such that x^n - x^2 - 1 is reducible. - _N. J. A. Sloane_, Jul 19 2005
%C Let M(n) be the n X n matrix m(i,j) = min(i,j); then the trace of M(n)^(-2) is a(n-1) = 6*n - 5. - _Benoit Cloitre_, Feb 09 2006
%C If Y is a 3-subset of an (2n+1)-set X then, for n >= 3, a(n-1) is the number of 3-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Dec 16 2007
%C All composite terms belong to A269345 as shown in there. - _Waldemar Puszkarz_, Apr 13 2016
%C First differences of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 23 2016
%C For b(n) = A103221(n) one has b(a(n)-1) = b(a(n)+1) = b(a(n)+2) = b(a(n)+3) = b(a(n)+4) = n+1 but b(a(n)) = n. So-called "dips" in A103221. See the Avner and Gross remark on p. 178. - _Wolfdieter Lang_, Sep 16 2016
%C A (n+1,n) pebbling move involves removing n + 1 pebbles from a vertex in a simple graph and placing n pebbles on an adjacent vertex. A two-player impartial (n+1,n) pebbling game involves two players alternating (n+1,n) pebbling moves. The first player unable to make a move loses. The sequence a(n) is also the minimum number of pebbles such that any assignment of those pebbles on a complete graph with 3 vertices is a next-player winning game in the two player impartial (k+1,k) pebbling game. These games are represented by A347637(3,n). - _Joe Miller_, Oct 18 2021
%C Interleaving of A017533 and A017605. - _Leo Tavares_, Nov 16 2021
%D Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
%H G. C. Greubel, <a href="/A016921/b016921.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%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="/A016921/a016921.jpg">Illustration: Hexagonal Lines</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 6*n + 1, n >= 0 (see the name).
%F G.f.: (1+5*x)/(1-x)^2.
%F A008615(a(n)) = n. - _Reinhard Zumkeller_, Feb 27 2008
%F A157176(a(n)) = A013730(n). - _Reinhard Zumkeller_, Feb 24 2009
%F a(n) = 4*(3*n-1) - a(n-1) (with a(0)=1). - _Vincenzo Librandi_, Nov 20 2010
%F E.g.f.: (1 + 6*x)*exp(x). - _G. C. Greubel_, Sep 18 2019
%F a(n) = A003215(n) - 6*A000217(n-1). See Hexagonal Lines illustration. - _Leo Tavares_, Sep 10 2021
%F From _Leo Tavares_, Oct 27 2021: (Start)
%F a(n) = 6*A001477(n-1) + 7
%F a(n) = A016813(n) + 2*A001477(n)
%F a(n) = A017605(n-1) + A008588(n-1)
%F a(n) = A016933(n) - 1
%F a(n) = A008588(n) + 1. (End)
%F Sum_{n>=0} (-1)^n/a(n) = Pi/6 + sqrt(3)*arccoth(sqrt(3))/3. - _Amiram Eldar_, Dec 10 2021
%e From _Ilya Gutkovskiy_, Apr 15 2016: (Start)
%e Illustration of initial terms:
%e o
%e o o o
%e o o o o
%e o o o o o o
%e o o o o o o o
%e o o o o o o o o o
%e o o o o o o o o o o
%e n=0 n=1 n=2 n=3
%e (End)
%p a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..56); # _Zerinvary Lajos_, Mar 16 2008
%t Range[1, 500, 6] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%o (Haskell)
%o a016921 = (+ 1) . (* 6)
%o a016921_list = [1, 7 ..] -- _Reinhard Zumkeller_, Jan 15 2013
%o (PARI) a(n)=6*n+1 \\ _Charles R Greathouse IV_, Mar 22 2016
%o (Python) for n in range(0,10**5):print(6*n+1) # _Soumil Mandal_, Apr 14 2016
%o (Magma) [6*n+1: n in [0..60]]; // _G. C. Greubel_, Sep 18 2019
%o (GAP) List([0..60], n-> 6*n+1); # _G. C. Greubel_, Sep 18 2019
%Y Cf. A093563 ((6, 1) Pascal, column m=1).
%Y Cf. A000567 (partial sums), A002476 (primes), A005408, A008588, A016813, A016933, A016945, A016957, A017281, A017533, A128470, A158057, A161700, A161705, A161709, A161714.
%Y a(n) = A007310(2*(n+1)); complement of A016969 with respect to A007310.
%Y Cf. A287326 (second column).
%Y Cf. A000217, A003215.
%Y Cf. A001477, A017605.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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