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A109613 Odd numbers repeated. 81

%I #149 Feb 08 2024 01:41:10

%S 1,1,3,3,5,5,7,7,9,9,11,11,13,13,15,15,17,17,19,19,21,21,23,23,25,25,

%T 27,27,29,29,31,31,33,33,35,35,37,37,39,39,41,41,43,43,45,45,47,47,49,

%U 49,51,51,53,53,55,55,57,57,59,59,61,61,63,63,65,65,67,67,69,69,71,71,73

%N Odd numbers repeated.

%C The number of rounds in a round-robin tournament with n competitors. - _A. Timothy Royappa_, Aug 13 2011

%C Diagonal sums of number triangle A113126. - _Paul Barry_, Oct 14 2005

%C When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - _Tanya Khovanova_, Apr 06 2008

%C The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - _Johannes W. Meijer_, Nov 12 2009

%C From _Reinhard Zumkeller_, Dec 05 2009: (Start)

%C First differences: A010673; partial sums: A000982;

%C A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);

%C A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);

%C A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);

%C A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)

%C Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - _Michael Somos_, May 29 2013

%C For n > 4: a(n) = A230584(n) - A230584(n-2). - _Reinhard Zumkeller_, Feb 10 2015

%C The arithmetic function v+-(n,2) as defined in A290988. - _Robert Price_, Aug 22 2017

%C For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - _Eric W. Weisstein_, Nov 17 2017

%C a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - _Wolfdieter Lang_, Feb 19 2020

%C Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - _Peter Bala_, Aug 30 2022

%C The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - _R. J. Mathar_, Feb 25 2023

%D Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).

%H Charles R Greathouse IV, <a href="/A109613/b109613.txt">Table of n, a(n) for n = 0..10000</a>

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

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

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Round-robin_tournament">Round-robin tournament</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(n) = 2*floor(n/2) + 1.

%F a(n) = A052928(n) + 1 = 2*A004526(n) + 1.

%F a(n) = A028242(n) + A110654(n).

%F a(n) = A052938(n-2) + A084964(n-2) for n > 1. - _Reinhard Zumkeller_, Aug 27 2005

%F G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - _Paul Barry_, Oct 14 2005

%F a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - _Philippe Deléham_, Nov 03 2008

%F a(n) = A001477(n) + A059841(n). - _Philippe Deléham_, Mar 31 2009

%F a(n) = 2*n - a(n-1), with a(0) = 1. - _Vincenzo Librandi_, Nov 13 2010

%F a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - _Peter Bala_, May 01 2012

%F a(n) = A182579(n+1, n). - _Reinhard Zumkeller_, May 06 2012

%F G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - _R. J. Mathar_, Jul 12 2016

%F E.g.f.: x*exp(x) + cosh(x). - _Ilya Gutkovskiy_, Jul 12 2016

%F From _Guenther Schrack_, Sep 10 2018: (Start)

%F a(-n) = -a(n-1).

%F a(n) = A047270(n+1) - (2*n + 2).

%F a(n) = A005408(A004526(n)). (End)

%F a(n) = A000217(n) / A004526(n+1), n > 0. - _Torlach Rush_, Nov 10 2023

%e G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...

%e Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - _Wolfdieter Lang_, Feb 19 2020

%p A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # _Wesley Ivan Hurt_, Oct 22 2013

%t Flatten@ Array[{2# - 1, 2# - 1} &, 37] (* _Robert G. Wilson v_, Jul 07 2012 *)

%t (# - Boole[EvenQ[#]] &) /@ Range[80] (* _Alonso del Arte_, Sep 11 2019 *)

%t With[{c=2*Range[0,40]+1},Riffle[c,c]] (* _Harvey P. Dale_, Jan 02 2020 *)

%o (Haskell)

%o a109613 = (+ 1) . (* 2) . (`div` 2)

%o a109613_list = 1 : 1 : map (+ 2) a109613_list

%o -- _Reinhard Zumkeller_, Oct 27 2012, Feb 21 2011

%o (PARI) A109613(n)=n>>1<<1+1 \\ _Charles R Greathouse IV_, Feb 24 2011

%o (Sage) def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # _Michael Somos_, May 29 2013

%o (Scala) ((1 to 49) by 2) flatMap { List.fill(2)(_) } // _Alonso del Arte_, Sep 11 2019

%Y Cf. A000217, A063196, A110660, A186421, A186422, A211955, A230584, A290988.

%Y Complement of A052928 with respect to the universe A004526. - _Guenther Schrack_, Aug 21 2018

%Y First differences of A000982, A061925, A074148, A105343, A116940, and A179207. - _Guenther Schrack_, Aug 21 2018

%K nonn,easy

%O 0,3

%A _Reinhard Zumkeller_, Aug 01 2005

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)