|
%I #379 Apr 22 2024 07:20:56
%S 0,2,8,18,32,50,72,98,128,162,200,242,288,338,392,450,512,578,648,722,
%T 800,882,968,1058,1152,1250,1352,1458,1568,1682,1800,1922,2048,2178,
%U 2312,2450,2592,2738,2888,3042,3200,3362,3528,3698,3872,4050,4232,4418
%N a(n) = 2*n^2.
%C Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - _Roberto E. Martinez II_, Jan 07 2002
%C "If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature, Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.
%C Let z(1) = i = sqrt(-1), z(k+1) = 1/(z(k)+2i); then a(n) = (-1)*Imag(z(n+1))/Real(z(n+1)). - _Benoit Cloitre_, Aug 06 2002
%C Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - _Jeremy Gardiner_, Dec 19 2004
%C Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2, (15+21)/2, ... . - _Amarnath Murthy_, Aug 05 2005
%C These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010
%C Integral areas of isosceles right triangles with rational legs (legs are 2n and triangles are nondegenerate for n > 0). - _Rick L. Shepherd_, Sep 29 2009
%C Even squares divided by 2. - _Omar E. Pol_, Aug 18 2011
%C Number of stars when distributed as in the U.S.A. flag: n rows with n+1 stars and, between each pair of these, one row with n stars (i.e., n-1 of these), i.e., n*(n+1)+(n-1)*n = 2*n^2 = A001105(n). - _César Eliud Lozada_, Sep 17 2012
%C Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak having height n-3. - _David Scambler_, Apr 29 2013
%C Sum of the partition parts of 2n into exactly two parts. - _Wesley Ivan Hurt_, Jun 01 2013
%C Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective odd leg a (A180620); sequence gives values c-a, sorted with duplicates removed. - _K. G. Stier_, Nov 04 2013
%C Number of roots in the root systems of type B_n and C_n (for n > 1). - _Tom Edgar_, Nov 05 2013
%C Area of a square with diagonal 2n. - _Wesley Ivan Hurt_, Jun 18 2014
%C This sequence appears also as the first and second member of the quartet [a(n), a(n), p(n), p(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p(n) = A046092(n). See an Oct 15 2014 comment on A147973 where also a reference is given. - _Wolfdieter Lang_, Oct 16 2014
%C a(n) are the only integers m where (A000005(m) + A000203(m)) = (number of divisors of m + sum of divisors of m) is an odd number. - _Richard R. Forberg_, Jan 09 2015
%C a(n) represents the first term in a sum of consecutive integers running to a(n+1)-1 that equals (2n+1)^3. - _Patrick J. McNab_, Dec 24 2016
%C Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - _Eric W. Weisstein_, Jul 29 2017
%C Also the Wiener index of the n-cocktail party graph for n > 1. - _Eric W. Weisstein_, Sep 07 2017
%C Numbers represented as the palindrome 242 in number base B including B=2 (binary), 3 (ternary) and 4: 242(2)=18, 242(3)=32, 242(4)=50, ... 242(9)=200, 242(10)=242, ... - _Ron Knott_, Nov 14 2017
%C a(n) is the square of the hypotenuse of an isosceles right triangle whose sides are equal to n. - _Thomas M. Green_, Aug 20 2019
%C The sequence contains all odd powers of 2 (A004171) but no even power of 2 (A000302). - _Torlach Rush_, Oct 10 2019
%C From _Bernard Schott_, Aug 31 2021 and Sep 16 2021: (Start)
%C Apart from 0, integers such that the number of even divisors (A183063) is odd.
%C Proof: every n = 2^q * (2k+1), q, k >= 0, then 2*n^2 = 2^(2q+1) * (2k+1)^2; now, gcd(2, 2k+1) = 1, tau(2^(2q+1)) = 2q+2 and tau((2k+1)^2) = 2u+1 because (2k+1)^2 is square, so, tau(2*n^2) = (2q+2) * (2u+1).
%C The 2q+2 divisors of 2^(2q+1) are {1, 2, 2^2, 2^3, ..., 2^(2q+1)}, so 2^(2q+1) has 2q+1 even divisors {2^1, 2^2, 2^3, ..., 2^(2q+1)}.
%C Conclusion: these 2q+1 even divisors create with the 2u+1 odd divisors of (2k+1)^2 exactly (2q+1)*(2u+1) even divisors of 2*n^2, and (2q+1)*(2u+1) is odd. (End)
%C a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - _Paul Weisenhorn_, Jan 29 2022
%C Number of points at L1 distance = 2 from any given point in Z^n. - _Shel Kaphan_, Feb 25 2023
%D Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
%D Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles, Paradoxes and Problems, Chapter 2 entitled "The Calculus of Finite Differences," W. W. Norton and Company, New York, 2001, pages 12-13.
%D L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.
%D Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000, p. 213.
%H Vincenzo Librandi, <a href="/A001105/b001105.txt">Table of n, a(n) for n = 0..1000</a>
%H Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
%H Vladimir Ladma, <a href="http://www.traced-ideas.eu/atom/atomcore.html">Magic Numbers</a>.
%H Vladimir Pletser, <a href="http://arxiv.org/abs/1501.06098">General solutions of sums of consecutive cubed integers equal to squared integers</a>, arXiv:1501.06098 [math.NT], 2015.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv:1406.3081 [math.CO], 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (-1)^(n+1) * A053120(2*n, 2).
%F G.f.: 2*x*(1+x)/(1-x)^3.
%F a(n) = A100345(n, n).
%F Sum_{n>=1} 1/a(n) = Pi^2/12 =A013661/2. [Jolley eq. 319]. - _Gary W. Adamson_, Dec 21 2006
%F a(n) = A049452(n) - A033991(n). - _Zerinvary Lajos_, Jun 12 2007
%F a(n) = A016742(n)/2. - _Zerinvary Lajos_, Jun 20 2008
%F a(n) = 2 * A000290(n). - _Omar E. Pol_, May 14 2008
%F a(n) = 4*n + a(n-1) - 2, n > 0. - _Vincenzo Librandi_
%F a(n) = A002378(n-1) + A002378(n). - Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010 [Corrected by _Klaus Purath_, Jun 18 2020]
%F a(n) = A176271(n,k) + A176271(n,n-k+1), 1 <= k <= n. - _Reinhard Zumkeller_, Apr 13 2010
%F a(n) = A007607(A000290(n)). - _Reinhard Zumkeller_, Feb 12 2011
%F For n > 0, a(n) = 1/coefficient of x^2 in the Maclaurin expansion of 1/(cos(x)+n-1). - _Francesco Daddi_, Aug 04 2011
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Artur Jasinski_, Nov 24 2011
%F a(n) = A070216(n,n) for n > 0. - _Reinhard Zumkeller_, Nov 11 2012
%F a(n) = A014132(2*n-1,n) for n > 0. - _Reinhard Zumkeller_, Dec 12 2012
%F a(n) = A000217(n) + A000326(n). - _Omar E. Pol_, Jan 11 2013
%F (a(n) - A000217(k))^2 = A000217(2*n-1-k)*A000217(2*n+k) + n^2, for all k. - _Charlie Marion_, May 04 2013
%F a(n) = floor(1/(1-cos(1/n))), n > 0. - _Clark Kimberling_, Oct 08 2014
%F a(n) = A251599(3*n-1) for n > 0. - _Reinhard Zumkeller_, Dec 13 2014
%F a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - _Wesley Ivan Hurt_, Mar 12 2015
%F a(n) = A002061(n+1) + A165900(n). - _Torlach Rush_, Feb 21 2019
%F E.g.f.: 2*exp(x)*x*(1 + x). - _Stefano Spezia_, Oct 12 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - _Amiram Eldar_, Jul 03 2020
%F From _Amiram Eldar_, Feb 03 2021: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = sqrt(2)*sinh(Pi/sqrt(2))/Pi.
%F Product_{n>=1} (1 - 1/a(n)) = sqrt(2)*sin(Pi/sqrt(2))/Pi. (End)
%e a(3) = 18; since 2(3) = 6 has 3 partitions with exactly two parts: (5,1), (4,2), (3,3). Adding all the parts, we get: 1 + 2 + 3 + 3 + 4 + 5 = 18. - _Wesley Ivan Hurt_, Jun 01 2013
%p A001105:=n->2*n^2; seq(A001105(k), k=0..100); # _Wesley Ivan Hurt_, Oct 29 2013
%t 2 Range[0, 50]^2 (* _Harvey P. Dale_, Jan 23 2011 *)
%t LinearRecurrence[{3, -3, 1}, {2, 8, 18}, {0, 20}] (* _Eric W. Weisstein_, Jul 28 2017 *)
%t 2 PolygonalNumber[4, Range[0, 20]] (* _Eric W. Weisstein_, Jul 28 2017 *)
%o (Magma) [2*n^2: n in [0..50] ]; // _Vincenzo Librandi_, Apr 30 2011
%o (PARI) a(n) = 2*n^2; \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Haskell)
%o a001105 = a005843 . a000290 -- _Reinhard Zumkeller_, Dec 12 2012
%o (Sage) [2*n^2 for n in (0..20)] # _G. C. Greubel_, Feb 22 2019
%o (GAP) List([0..50],n->2*n^2); # _Muniru A Asiru_, Feb 24 2019
%Y Cf. A000290, A006331 (partial sums), A016742, A056106, A116471, A245508, A251599, A002061, A165900.
%Y Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488.
%Y Cf. A058331 and A247375. - _Bruno Berselli_, Sep 16 2014
%Y Cf. A194715 (4-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - _Eric W. Weisstein_, Jul 29 2017
%Y Cf. A139098, A077591.
%Y Cf. A000217, A002266.
%Y Integers such that: this sequence (the number of even divisors is odd), A028982 (the number of odd divisors is odd), A028983 (the number of odd divisors is even), A183300 (the number of even divisors is even).
%K nonn,easy,changed
%O 0,2
%A Bernd.Walter(AT)frankfurt.netsurf.de
| 