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 A001105 a(n) = 2*n^2. 210
 0, 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882, 968, 1058, 1152, 1250, 1352, 1458, 1568, 1682, 1800, 1922, 2048, 2178, 2312, 2450, 2592, 2738, 2888, 3042, 3200, 3362, 3528, 3698, 3872, 4050, 4232, 4418 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - Roberto E. Martinez II, Jan 07 2002 "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. Let z(1) = I; (I^2 = -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 Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - Jeremy Gardiner, Dec 19 2004 Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2,(15+21)/2, ... . - Amarnath Murthy, Aug 05 2005 These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010 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 Even squares divided by 2. - Omar E. Pol, Aug 18 2011 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 Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak has height n-3. - David Scambler, Apr 29 2013 Sum of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 01 2013 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 Number of roots in the root systems of type B_n and C_n (for n>1). - Tom Edgar, Nov 05 2013 Area of a square with diagonal 2n. - Wesley Ivan Hurt, Jun 18 2014 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 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 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 Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - Eric W. Weisstein, Jul 29 2017 Also the Wiener index of the n-cocktail party graph for n > 1. - Eric W. Weisstein, Sep 07 2017 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 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 The sequence contains all odd powers of 2 (A004171) but no even power of 2 (A000302). - Torlach Rush, Oct 10 2019 From Bernard Schott, Aug 31 2021 and Sep 16 2021: (Start) Apart from 0, integers such that the number of even divisors (A183063) is odd. 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). 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)}. 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) a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - Paul Weisenhorn, Jan 29 2022 REFERENCES Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973. 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. L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44. Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000, p. 213. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36. Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8. Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019. Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. - N. J. A. Sloane, Feb 13 2013 Vladimir Ladma, Magic Numbers. Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015. Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014. Eric Weisstein's World of Mathematics, Cocktail Party Graph. Eric Weisstein's World of Mathematics, Graph Cycle. Eric Weisstein's World of Mathematics, Wiener Index. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = (-1)^(n+1) * A053120(2*n, 2). G.f.: 2*x*(1+x)/(1-x)^3. a(n) = A100345(n, n). Sum_{n>=1} 1/a(n) = Pi^2/12 [Jolley eq. 319]. - Gary W. Adamson, Dec 21 2006 a(n) = A049452(n) - A033991(n). - Zerinvary Lajos, Jun 12 2007 a(n) = A016742(n)/2. - Zerinvary Lajos, Jun 20 2008 a(n) = 2 * A000290(n). - Omar E. Pol, May 14 2008 a(n) = 4*n + a(n-1) - 2, n > 0. - Vincenzo Librandi a(n) = A002378(n-1) + A002378(n). - Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010 [Corrected by Klaus Purath, Jun 18 2020] a(n) = A176271(n,k) + A176271(n,n-k+1), 1 <= k <= n. - Reinhard Zumkeller, Apr 13 2010 a(n) = A007607(A000290(n)). - Reinhard Zumkeller, Feb 12 2011 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 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Artur Jasinski, Nov 24 2011 a(n) = A070216(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012 a(n) = A014132(2*n-1,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012 a(n) = A000217(n) + A000326(n). - Omar E. Pol, Jan 11 2013 (a(n) - A000217(k))^2  = A000217(2*n-1-k)*A000217(2*n+k) + n^2, for all k. - Charlie Marion, May 04 2013 a(n) = floor(1/(1-cos(1/n))), n > 0. - Clark Kimberling, Oct 08 2014 a(n) = A251599(3*n-1) for n > 0. - Reinhard Zumkeller, Dec 13 2014 a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - Wesley Ivan Hurt, Mar 12 2015 a(n) = A002061(n+1) + A165900(n). - Torlach Rush, Feb 21 2019 E.g.f.: 2*exp(x)*x*(1 + x). - Stefano Spezia, Oct 12 2019 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - Amiram Eldar, Jul 03 2020 From Amiram Eldar, Feb 03 2021: (Start) Product_{n>=1} (1 + 1/a(n)) = sqrt(2)*sinh(Pi/sqrt(2))/Pi. Product_{n>=1} (1 - 1/a(n)) = sqrt(2)*sin(Pi/sqrt(2))/Pi. (End) EXAMPLE 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 MAPLE A001105:=n->2*n^2; seq(A001105(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013 MATHEMATICA 2 Range[0, 50]^2 (* Harvey P. Dale, Jan 23 2011 *) LinearRecurrence[{3, -3, 1}, {2, 8, 18}, {0, 20}] (* Eric W. Weisstein, Jul 28 2017 *) 2 PolygonalNumber[4, Range[0, 20]] (* Eric W. Weisstein, Jul 28 2017 *) PROG (Magma) [2*n^2: n in [0..50] ]; // Vincenzo Librandi, Apr 30 2011 (PARI) a(n) = 2*n^2; \\ Charles R Greathouse IV, Jun 16 2011 (Haskell) a001105 = a005843 . a000290  -- Reinhard Zumkeller, Dec 12 2012 (Sage) [2*n^2 for n in (0..20)] # G. C. Greubel, Feb 22 2019 (GAP) List([0..50], n->2*n^2); # Muniru A Asiru, Feb 24 2019 CROSSREFS Cf. A000290, A006331 (partial sums), A016742, A056106, A116471, A245508, A251599, A002061, A165900. Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488. Cf. A058331 and A247375. - Bruno Berselli, Sep 16 2014 Cf. A194715 (4-cycles in the triangular honeycomb obtuse knight graph), A290391 (5-cycles), A290392 (6-cycles). - Eric W. Weisstein, Jul 29 2017 Cf. A139098, A077591. Cf. A000217, A002266. 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). Sequence in context: A067051 A074629 A209303 * A336489 A051787 A050804 Adjacent sequences:  A001102 A001103 A001104 * A001106 A001107 A001108 KEYWORD nonn,easy AUTHOR Bernd.Walter(AT)frankfurt.netsurf.de STATUS approved

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Last modified October 2 16:01 EDT 2022. Contains 357226 sequences. (Running on oeis4.)