cp's OEIS Frontend

A001105 a(n) = 2*n^2.

%I A001105 #406 Aug 04 2025 18:03:55
%S A001105 0,2,8,18,32,50,72,98,128,162,200,242,288,338,392,450,512,578,648,722,
%T A001105 800,882,968,1058,1152,1250,1352,1458,1568,1682,1800,1922,2048,2178,
%U A001105 2312,2450,2592,2738,2888,3042,3200,3362,3528,3698,3872,4050,4232,4418
%N A001105 a(n) = 2*n^2.
%C A001105 Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - _Roberto E. Martinez II_, Jan 07 2002
%C A001105 "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 A001105 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 A001105 Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - _Jeremy Gardiner_, Dec 19 2004
%C A001105 Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2, (15+21)/2, ... . - _Amarnath Murthy_, Aug 05 2005
%C A001105 These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010
%C A001105 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 A001105 Even squares divided by 2. - _Omar E. Pol_, Aug 18 2011
%C A001105 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 A001105 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 A001105 Sum of the partition parts of 2n into exactly two parts. - _Wesley Ivan Hurt_, Jun 01 2013
%C A001105 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 A001105 Number of roots in the root systems of type B_n and C_n (for n > 1). - _Tom Edgar_, Nov 05 2013
%C A001105 Area of a square with diagonal 2n. - _Wesley Ivan Hurt_, Jun 18 2014
%C A001105 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 A001105 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 A001105 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 A001105 Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - _Eric W. Weisstein_, Jul 29 2017
%C A001105 Also the Wiener index of the n-cocktail party graph for n > 1. - _Eric W. Weisstein_, Sep 07 2017
%C A001105 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 A001105 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 A001105 The sequence contains all odd powers of 2 (A004171) but no even power of 2 (A000302). - _Torlach Rush_, Oct 10 2019
%C A001105 From _Bernard Schott_, Aug 31 2021 and Sep 16 2021: (Start)
%C A001105 Apart from 0, integers such that the number of even divisors (A183063) is odd.
%C A001105 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 A001105 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 A001105 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 A001105 a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - _Paul Weisenhorn_, Jan 29 2022
%C A001105 Number of points at L1 distance = 2 from any given point in Z^n. - _Shel Kaphan_, Feb 25 2023
%C A001105 Integer that multiplies (h^2)/(m*L^2) to give the energy of a 1-D quantum mechanical particle in a box whenever it is an integer multiple of (h^2)/(m*L^2), where h = Planck's constant, m = mass of particle, and L = length of box. - _A. Timothy Royappa_, Mar 14 2025
%D A001105 Peter Atkins, Julio De Paula, and James Keeler, "Atkins' Physical Chemistry," Oxford University Press, 2023, p. 31.
%D A001105 Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
%D A001105 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 A001105 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.
%D A001105 Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000, p. 213.
%H A001105 Vincenzo Librandi, <a href="/A001105/b001105.txt">Table of n, a(n) for n = 0..1000</a>
%H A001105 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 A001105 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 A001105 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A001105 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 A001105 Vladimir Ladma, <a href="http://www.traced-ideas.eu/atom/atomcore.html">Magic Numbers</a>.
%H A001105 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 A001105 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 A001105 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H A001105 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H A001105 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H A001105 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A001105 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A001105 a(n) = (-1)^(n+1) * A053120(2*n, 2).
%F A001105 G.f.: 2*x*(1+x)/(1-x)^3.
%F A001105 a(n) = A100345(n, n).
%F A001105 Sum_{n>=1} 1/a(n) = Pi^2/12 =A072691. [Jolley eq. 319]. - _Gary W. Adamson_, Dec 21 2006
%F A001105 a(n) = A049452(n) - A033991(n). - _Zerinvary Lajos_, Jun 12 2007
%F A001105 a(n) = A016742(n)/2. - _Zerinvary Lajos_, Jun 20 2008
%F A001105 a(n) = 2 * A000290(n). - _Omar E. Pol_, May 14 2008
%F A001105 a(n) = 4*n + a(n-1) - 2, n > 0. - _Vincenzo Librandi_
%F A001105 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 A001105 a(n) = A176271(n,k) + A176271(n,n-k+1), 1 <= k <= n. - _Reinhard Zumkeller_, Apr 13 2010
%F A001105 a(n) = A007607(A000290(n)). - _Reinhard Zumkeller_, Feb 12 2011
%F A001105 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 A001105 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Artur Jasinski_, Nov 24 2011
%F A001105 a(n) = A070216(n,n) for n > 0. - _Reinhard Zumkeller_, Nov 11 2012
%F A001105 a(n) = A014132(2*n-1,n) for n > 0. - _Reinhard Zumkeller_, Dec 12 2012
%F A001105 a(n) = A000217(n) + A000326(n). - _Omar E. Pol_, Jan 11 2013
%F A001105 (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 A001105 a(n) = floor(1/(1-cos(1/n))), n > 0. - _Clark Kimberling_, Oct 08 2014
%F A001105 a(n) = A251599(3*n-1) for n > 0. - _Reinhard Zumkeller_, Dec 13 2014
%F A001105 a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - _Wesley Ivan Hurt_, Mar 12 2015
%F A001105 a(n) = A002061(n+1) + A165900(n). - _Torlach Rush_, Feb 21 2019
%F A001105 E.g.f.: 2*exp(x)*x*(1 + x). - _Stefano Spezia_, Oct 12 2019
%F A001105 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - _Amiram Eldar_, Jul 03 2020
%F A001105 From _Amiram Eldar_, Feb 03 2021: (Start)
%F A001105 Product_{n>=1} (1 + 1/a(n)) = sqrt(2)*sinh(Pi/sqrt(2))/Pi.
%F A001105 Product_{n>=1} (1 - 1/a(n)) = sqrt(2)*sin(Pi/sqrt(2))/Pi. (End)
%e A001105 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 A001105:=n->2*n^2; seq(A001105(k), k=0..100); # _Wesley Ivan Hurt_, Oct 29 2013
%t A001105 2 Range[0, 50]^2 (* _Harvey P. Dale_, Jan 23 2011 *)
%t A001105 LinearRecurrence[{3, -3, 1}, {2, 8, 18}, {0, 20}] (* _Eric W. Weisstein_, Jul 28 2017 *)
%t A001105 2 PolygonalNumber[4, Range[0, 20]] (* _Eric W. Weisstein_, Jul 28 2017 *)
%o A001105 (Magma) [2*n^2: n in [0..50] ]; // _Vincenzo Librandi_, Apr 30 2011
%o A001105 (PARI) a(n) = 2*n^2; \\ _Charles R Greathouse IV_, Jun 16 2011
%o A001105 (Haskell)
%o A001105 a001105 = a005843 . a000290  -- _Reinhard Zumkeller_, Dec 12 2012
%o A001105 (Sage) [2*n^2 for n in (0..20)] # _G. C. Greubel_, Feb 22 2019
%o A001105 (GAP) List([0..50],n->2*n^2); # _Muniru A Asiru_, Feb 24 2019
%o A001105 (Python)
%o A001105 def A001105(n): return n**2<<1 # _Chai Wah Wu_, Aug 04 2025
%Y A001105 Cf. A000290, A006331 (partial sums), A016742, A056106, A116471, A245508, A251599, A002061, A165900.
%Y A001105 Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.
%Y A001105 Cf. A058331 and A247375. - _Bruno Berselli_, Sep 16 2014
%Y A001105 Cf. A194715 (4-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - _Eric W. Weisstein_, Jul 29 2017
%Y A001105 Cf. A139098, A077591.
%Y A001105 Cf. A000217, A002266.
%Y A001105 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 A001105 nonn,easy
%O A001105 0,2
%A A001105 Bernd.Walter(AT)frankfurt.netsurf.de