%I A046092
%S A046092 0,4,12,24,40,60,84,112,144,180,220,264,312,364,420,480,544,612,684,760,
%T A046092 840,924,1012,1104,1200,1300,1404,1512,1624,1740,1860,1984,2112,2244,
%U A046092 2380,2520,2664,2812,2964,3120,3280,3444,3612,3784,3960,4140,4324
%N A046092 2n(n+1).
%C A046092 Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; 
               sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), 
               Z values are A001844.
%C A046092 In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given 
               by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408) 
               and x^2 over the odd squares (A016754). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 11 2004
%C A046092 a(n) = number of edges in (n+1) X (n+1) square grid with all horizontal 
               and vertical segments filled in - Asher Auel (asher.auel(AT)reed.edu) 
               Jan 12, 2000.
%C A046092 a(n) is the only number satisfying an inequality related to zeta(2) and 
               zeta(3): sum(i>a(n)+1,1/i^2) < sum(i>n,1/i^3) < sum(i>a(n),1/i^2). 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 02 2001
%C A046092 Number of right triangles made from vertices of a regular n-gon when 
               n is even - Sen-Peng You (giawgwan(AT)single.url.com.tw), Apr 05 
               2001
%C A046092 Number of ways to change two non-identical letters in the word aabbccdd..., 
               where there are n type of letters. - Zerinvary Lajos (zlaja(AT)freemail.hu), 
               Feb 15 2005
%C A046092 a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional 
               hypercube (e.g. squares have 4 corners, cubes have 12 edges, etc.). 
               - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005
%C A046092 Comments from Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: 
               "Consider a triangle, a pentagon, an eptagon, ..., a k-gon where 
               k is odd. We label a triangle with n=1, a pentagon with n=2, .., 
               a k-gon with n = floor(k/2). Imagine every player standing on every 
               vertex of the k-gon.
%C A046092 "Initially there are 2 frisbees on two neighboring players. Every time 
               they throw the frisbee to their neighbor with equal probability. 
               Then a(n) gives the average number of steps needed so that the frisbees 
               meet.
%C A046092 "I verified it by simulating the processes with a computer program. For 
               example a(2) = 12 because in a pentagon that's the expected number 
               of trials we need to perform. That is an exercise in Concrete Mathematics 
               and it can be done using generating functions."
%C A046092 First difference of a(n) is 4n = A008586(n). Any entry k of the sequence 
               is followed by k + 2*{1 + sqrt(2k + 1)}. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 04 2006
%C A046092 A diagonal of A059056. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 18 2007
%C A046092 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n-1) 
               is equal to the number of 2-subsets of X containing none of X_i, 
               (i=1,...n). - Milan R. Janjic (agnus(AT)blic.net), Jul 16 2007
%C A046092 Sequence allows us to find X values of the equation: 2*X^3 + X^2 = Y^2. 
               To find Y values: b(n)=2n(n+1)(2n+1). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               Nov 06 2007
%C A046092 Number of (n+1)-permutations of 3 objects u,v,w, with repetition allowed, 
               containing n-1 u's. Example: a(1)=4 because we have vv, vw, wv and 
               ww; a(2)=12 because we can place u in each of the previous four 2-permutations 
               either in front, or in the middle, or at the end. - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Dec 27 2007
%C A046092 Sequence found by reading the line from 0, in the direction 0, 4,... 
               and the same line from 0, in the direction 0, 12,..., in the square 
               spiral whose vertices are the triangular numbers A000217. - Omar 
               E. Pol (info(AT)polprimos.com), May 03 2008
%C A046092 Twice oblong numbers. - Omar E. Pol (info(AT)polprimos.com), May 03 2008
%C A046092 a(n) is also the least weight of self-conjugate partitions having n different 
               even parts. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 
               18 2008]
%C A046092 Except for the two terms of [A141530] and the first term of [A046092[, 
               if X=[A141530], A=[A078371], Y=[A046092], we have, for all others 
               terms, Pell's equation: X^2-A*Y^2=1. Example: 9^2-5*4^2=1; 55^2-21*12^2=1; 
               161^2-45*24^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 13 2009]
%C A046092 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start)
%C A046092 The general formula for alternating sums of powers of even integers is 
               in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k 
               P(n,2k+1))/2. Here n=2, thus
%C A046092 a(k) = |(P(2,1)-(-1)^k P(2,2k+1))/2|. (End)
%C A046092 The sum of squares of n+1 consecutive numbers between a(n)-n and a(n) 
               inclusive equals the sum of squares of n consecutive numbers following 
               a(n). For example, for n = 2, a(2) = 12, and the corresponding equation 
               is 10^2+11^2+12^2=13^2+14^2. [From Tanya Khovanova (tanyakh(AT)yahoo.com), 
               Jul 20 2009]
%C A046092 Number of units of a(n) belongs to a periodic sequence: 0, 4, 2, 4, 0. 
               [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
%D A046092 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 3.
%D A046092 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, 
               p. 125, 1964.
%D A046092 Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, 
               Reading, Massachusetts: Addison-Wesley, 1994.
%D A046092 A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete 
               Math., 308 (2008), 2492-2501. [From Augustine O. Munagi (amunagi(AT)yahoo.com), 
               Dec 18 2008]
%H A046092 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A046092 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/
               pythag.html">Pythagorean Triples and Online Calculators</a>
%H A046092 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PythagoreanTriple.html">Link to a section of The World of Mathematics.</
               a>
%H A046092 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AztecDiamond.html">Aztec Diamond</a>
%H A046092 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GearGraph.html">Gear Graph</a>
%H A046092 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HamiltonianPath.html">Hamiltonian Path</a>
%H A046092 O. E. Pol, <a href="http://polprimos.com">Determinacion geometrica de 
               los numeros primos y perfectos</a>.
%H A046092 <a href="http://blog.tanyakhovanova.com/?p=151">A Miracle Equation</a> 
               [From Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 20 2009]
%F A046092 a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               May 25 2004
%F A046092 a(n) = C(2n, 2) - n = 4*C(n, 2) - Zerinvary Lajos (zlaja(AT)freemail.hu), 
               Feb 15 2005
%F A046092 a(n)=A028896-A002378; a(n)=A124080-A028896; a(n)=A049598-A033996. - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 06 2007
%F A046092 Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol 
               (info(AT)polprimos.com), May 03 2008
%F A046092 a(n)=a(n-1)+4n O.g.f.:4x/(1-x)^3 E.g.f.:Exp(x)*(2x^2+4x) [From Geoffrey 
               Critzer (critzer.geoffrey(AT)usd443.org), May 17 2009]
%F A046092 Contribution from Stephen Crowley (crow(AT)crowlogic.net), Jul 26 2009: 
               (Start)
%F A046092 a(n)=1/int(-(x*n+x-1)*(step((-1+x*n)/n)-1)*n*step((x*n+x-1)/(n+1)),x=0..1) 
               where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function
%F A046092 sum(1/a(n),n=1..inf)=sum(1/((2*n)*(n+1)),n=1..inf)=1/2 (End)
%F A046092 a(n)=4*n+a(n-1)-4 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A046092 a(7)=112 because 112 = 2*7*(7+1).
%e A046092 The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25),...
%e A046092 The first such partitions, corresponding to a(n)=1,2,3,4, are 2+2, 4+4+2+2, 
               6+6+4+4+2+2, 8+8+6+6+4+4+2+2. [From Augustine O. Munagi (amunagi(AT)yahoo.com), 
               Dec 18 2008]
%e A046092 For n=2, a(2)=4*2+0-4=4; n=3, a(3)=4*3+4-4=12; n=4, a(4)=4*4+12-4=24 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A046092 a:=n->sum(n+2*j, j=0..n): seq(a(n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 29 2007
%p A046092 a:=n->sum(n, k=0..n):seq(a(n)+sum(n, k=4..n), n=1..50); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 20 2008
%p A046092 with(finance):seq(add(futurevalue( k, 3, 2),k=0..n)/4,n=0..46); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008
%p A046092 with(finance):seq(add(cashflows([n,k,k], 0 ),k=0..n),n=0..51); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008
%t A046092 Table[2n(n + 1), {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 03 2006
%Y A046092 Cf. A045943, A028895.
%Y A046092 Cf. A002943, A054000, A000330, A007290.
%Y A046092 Main diagonal of array in A001477.
%Y A046092 a(n)=A100345(n+1, n-1) for n>0.
%Y A046092 Equals A033996/2
%Y A046092 Cf. A002378, A033996, A124080, A028896, A049598.
%Y A046092 Cf. A005563.
%Y A046092 Cf. A000217, A033586, A085250.
%Y A046092 Cf. A001844 [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 
               2008]
%Y A046092 Cf. A078371, A141530 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 13 2009]
%Y A046092 Sequence in context: A081937 A115228 A088557 this_sequence A008241 A008216 
               A008074
%Y A046092 Adjacent sequences: A046089 A046090 A046091 this_sequence A046093 A046094 
               A046095
%K A046092 nonn,easy,nice,new
%O A046092 0,2
%A A046092 N. J. A. Sloane (njas(AT)research.att.com), Eric Weisstein (eric(AT)weisstein.com)