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A046092 4 times triangular numbers: a(n) = 2*n*(n+1).

%I A046092 #317 Apr 29 2025 21:01:08
%S A046092 0,4,12,24,40,60,84,112,144,180,220,264,312,364,420,480,544,612,684,
%T A046092 760,840,924,1012,1104,1200,1300,1404,1512,1624,1740,1860,1984,2112,
%U A046092 2244,2380,2520,2664,2812,2964,3120,3280,3444,3612,3784,3960,4140,4324
%N A046092 4 times triangular numbers: a(n) = 2*n*(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_, Jun 11 2004
%C A046092 a(n) is the number of edges in n X n square grid with all horizontal and vertical segments filled in. - _Asher Auel_, Jan 12 2000 [Corrected by _Felix Huber_, Apr 09 2024]
%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_, Nov 02 2001
%C A046092 Number of right triangles made from vertices of a regular n-gon when n is even. - _Sen-Peng Eu_, 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_, 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 From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)
%C A046092 Consider a triangle, a pentagon, a heptagon, ..., 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 a player standing at each vertex of the k-gon.
%C A046092 Initially there are 2 frisbees, one held by each of two neighboring players. Every time they throw the frisbee to one of their two nearest neighbors with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.
%C A046092 I verified this 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. (End)
%C A046092 A diagonal of A059056. - _Zerinvary Lajos_, 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 Janjic_, Jul 16 2007
%C A046092 X values of solutions to the equation 2*X^3 + X^2 = Y^2. To find Y values: b(n) = 2n(n+1)(2n+1). - _Mohamed Bouhamida_, 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_, 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_, May 03 2008
%C A046092 a(n) is also the least weight of self-conjugate partitions having n different even parts. - _Augustine O. Munagi_, Dec 18 2008
%C A046092 From _Peter Luschny_, 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. - _Tanya Khovanova_, Jul 20 2009
%C A046092 Number of roots in the root system of type D_{n+1} (for n>2). - _Tom Edgar_, Nov 05 2013
%C A046092 Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of intersections of these ellipses (cf. A051890, A001844); a(n) = A051890(n+1) - 2 = A001844(n) - 1. - _Jaroslav Krizek_, Dec 27 2013
%C A046092 a(n) appears also as the second member of the quartet [p0(n), a(n), p2(n), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p2(n) = A054000(n+1) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - _Wolfdieter Lang_, Oct 15 2014
%C A046092 a(n) appears also as the third and fourth member of the quartet [p0(n), p0(n), a(n), a(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p0(n) = A001105(n). - _Wolfdieter Lang_, Oct 16 2014
%C A046092 Consider two equal rectangles composed of unit squares. Then surround the 1st rectangle with 1-unit-wide layers to build larger rectangles, and surround the 2nd rectangle just to hide the previous layers. If r(n) and h(n) are the number of unit squares needed for n layers in the 1st case and the 2nd case, then for all rectangles, we have a(n) = r(n) - h(n) for n>=1. - _Michel Marcus_, Sep 28 2015
%C A046092 When greater than 4, a(n) is the perimeter of a Pythagorean triangle with an even short leg 2*n. - _Agola Kisira Odero_, Apr 26 2016
%C A046092 Also the number of minimum connected dominating sets in the (n+1)-cocktail party graph. - _Eric W. Weisstein_, Jun 29 2017
%C A046092 a(n+1) is the harmonic mean of A000384(n+2) and A014105(n+1). - _Bob Andriesse_, Apr 27 2019
%C A046092 Consider a circular cake from which wedges of equal center angle c are cut out in clockwise succession and turned around so that the bottom comes to the top. This goes on until the cake shows its initial surface again. An interesting case occurs if 360°/c is not an integer. Then, with n = floor(360°/c), the number of wedges which have to be cut out and turned equals a(n). (For the number of cutting line segments see A005408.) - According to Peter Winkler's book "Mathematical Mind-Benders", which presents the problem and its solution (see Winkler, pp. 111, 115) the problem seems to be of French origin but little is known about its history. - _Manfred Boergens_, Apr 05 2022
%C A046092 a(n-3) is the maximum irregularity over all maximal 2-degenerate graphs with n vertices.  The extremal graphs are 2-stars (K_2 joined to n-2 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - _Allan Bickle_, May 29 2023
%C A046092 Number of ways of placing a domino on a (n+1)X(n+1) board of squares. - _R. J. Mathar_, Apr 24 2024
%C A046092 The sequence terms are the exponents in the expansion of (1/(1 + x)) * Sum_{n >= 0} x^n * Product_{k = 1..n} (1 - x^(2*k-1))/(1 + x^(2*k+1)) = 1 - x^4 + x^12 - x^24 + x^40 - x^60 + - ... (Andrews and Berndt, Entry 9.3.3, p. 229). Cf. A153140. - _Peter Bala_, Feb 15 2025
%C A046092 Number of edges in an (n+1)-dimensional orthoplex. 2D orthoplexes (diamonds) have 4 edges, 3D orthoplexes (octahedrons) have 12 edges, 4D orthoplexes (16-cell) have 24 edges, and so on. - _Aaron Franke_, Mar 23 2025
%D A046092 George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005.
%D A046092 Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
%D A046092 Albert 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 Peter Winkler, Mathematical Mind-Benders, Wellesley, Massachusetts: A K Peters, 2007.
%H A046092 Vincenzo Librandi, <a href="/A046092/b046092.txt">Table of n, a(n) for n = 0..10000</a>
%H A046092 Allan Bickle and Zhongyuan Che, <a href="https://doi.org/10.1016/j.dam.2023.01.020">Irregularities of Maximal k-degenerate Graphs</a>, Discrete Applied Math. 331 (2023) 70-87.
%H A046092 Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H A046092 H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.
%H A046092 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, preprint, 2016.
%H A046092 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47.
%H A046092 Z. Janelidze, F. van Niekerk, and J. Viljoen, <a href="https://arxiv.org/abs/2502.00704">What is the maximal connected partial symmetry index of a connected graph of a given size?</a>, arXiv:2502.00704 [math.CO], 2025. See p. 3.
%H A046092 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A046092 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%H A046092 Tanya Khovanova, <a href="http://blog.tanyakhovanova.com/?p=151">A Miracle Equation</a>.
%H A046092 Augustine O. Munagi, <a href="http://dx.doi.org/10.1016/j.disc.2007.05.022">Pairing conjugate partitions by residue classes</a>, Discrete Math., 308 (2008), 2492-2501. [From _Augustine O. Munagi_, Dec 18 2008]
%H A046092 Enrique Navarrete and Daniel Orellana, <a href="https://arxiv.org/abs/1907.10023">Finding Prime Numbers as Fixed Points of Sequences</a>, arXiv:1907.10023 [math.NT], 2019.
%H A046092 Omar E. Pol, <a href="http://polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A046092 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H A046092 Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences (2019) Vol. 8, No. 10.
%H A046092 Rusliansyah D. Suprijanto, <a href="http://dx.doi.org/10.12988/ams.2014.4140">Observation on Sums of Powers of Integers Divisible by Four</a>, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.
%H A046092 Leo Tavares, <a href="/A046092/a046092.jpg">Illustration: Diamond Rows</a>
%H A046092 Herman Tulleken, <a href="https://www.researchgate.net/publication/333296614_Polyominoes">Polyominoes 2.2: How they fit together</a>, (2019).
%H A046092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AztecDiamond.html">Aztec Diamond</a>.
%H A046092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H A046092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>.
%H A046092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>.
%H A046092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.
%H A046092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>.
%H A046092 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A046092 a(n) = A100345(n+1, n-1) for n>0.
%F A046092 a(n) = 2*A002378(n) = 4*A000217(n). - _Lekraj Beedassy_, May 25 2004
%F A046092 a(n) = C(2n, 2) - n = 4*C(n, 2). - _Zerinvary Lajos_, Feb 15 2005
%F A046092 From _Lekraj Beedassy_, Jun 04 2006: (Start)
%F A046092 a(n) - a(n-1)=4*n.
%F A046092 Let k=a(n). Then a(n+1) = k + 2*(1 + sqrt(2k + 1)). (End)
%F A046092 Array read by rows: row n gives A033586(n), A085250(n+1). - _Omar E. Pol_, May 03 2008
%F A046092 O.g.f.:4*x/(1-x)^3; e.g.f.: exp(x)*(2*x^2+4*x). - _Geoffrey Critzer_, May 17 2009
%F A046092 From _Stephen Crowley_, 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_{n>=1} 1/a(n) = 1/2. (End)
%F A046092 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=12. - _Harvey P. Dale_, Jul 25 2011
%F A046092 For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} (sin(x))^(2*n-1)*(cos(x))^3). - _Francesco Daddi_, Aug 02 2011
%F A046092 a(n) = A001844(n) - 1. - _Omar E. Pol_, Oct 03 2011
%F A046092 (a(n) - A000217(k))^2  = A000217(2n-k)*A000217(2n+1+k) - (A002378(n) - A000217(k)), for all k. See also A001105. - _Charlie Marion_, May 09 2013
%F A046092 From _Ivan N. Ianakiev_, Aug 30 2013: (Start)
%F A046092 a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.
%F A046092 t(k)*a(n) + t(k-1)*a(n+1) = a((n+1)*(t(k)-t(k-1)-1)), where k>=2, n>=1, t(k)=A000217(k). (End)
%F A046092 a(n) = A245300(n,n). - _Reinhard Zumkeller_, Jul 17 2014
%F A046092 2*a(n)+1 = A016754(n) = A005408(n)^2, the odd squares. - _M. F. Hasler_, Oct 02 2014
%F A046092 Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 = A187832. - _Ilya Gutkovskiy_, Mar 16 2017
%F A046092 a(n) = lcm(2*n,2*n+2). - _Enrique Navarrete_, Aug 30 2017
%F A046092 a(n)*a(n+k) + k^2 = m^2 (a perfect square), n >= 1, k >= 0. - _Ezhilarasu Velayutham_, May 13 2019
%F A046092 From _Amiram Eldar_, Jan 29 2021: (Start)
%F A046092 Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/2)/(Pi/2).
%F A046092 Product_{n>=1} (1 - 1/a(n)) = -2*cos(sqrt(3)*Pi/2)/Pi. (End)
%F A046092 a(n) = A016754(n) - A001844(n). - _Leo Tavares_, Sep 20 2022
%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. - _Augustine O. Munagi_, Dec 18 2008
%t A046092 Table[2 n (n + 1), {n, 0, 50}] (* _Stefan Steinerberger_, Apr 03 2006 *)
%t A046092 LinearRecurrence[{3, -3, 1}, {0, 4, 12}, 50] (* _Harvey P. Dale_, Jul 25 2011 *)
%t A046092 4*Binomial[Range[50], 2] (* _Harvey P. Dale_, Jul 25 2011 *)
%o A046092 (PARI) a(n)=binomial(n+1,2)<<2 \\ _Charles R Greathouse IV_, Jun 10 2011
%o A046092 (Magma) [2*n*(n+1): n in [0..50]]; // _Vincenzo Librandi_, Oct 04 2011
%o A046092 (Maxima) A046092(n):=2*n*(n+1)$
%o A046092 makelist(A046092(n),n,0,30); /* _Martin Ettl_, Nov 08 2012 */
%o A046092 (Haskell)
%o A046092 a046092 = (* 2) . a002378  -- _Reinhard Zumkeller_, Dec 15 2013
%o A046092 (Python)
%o A046092 def A046092(n): return n*(n+1)<<1 # _Chai Wah Wu_, Mar 11 2025
%Y A046092 Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.
%Y A046092 Main diagonal of array in A001477.
%Y A046092 Equals A033996/2. Cf. A001844. - _Augustine O. Munagi_, Dec 18 2008
%Y A046092 Cf. A078371, A141530 (see Librandi's comment in A078371).
%Y A046092 Cf. A097080, A001845.
%Y A046092 Cf. similar sequences listed in A299645.
%Y A046092 Cf. A005408.
%Y A046092 Cf. A016754.
%Y A046092 Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
%K A046092 nonn,easy,nice
%O A046092 0,2
%A A046092 _N. J. A. Sloane_ and _Eric W. Weisstein_