

A046092


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


180



0, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324
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OFFSET

0,2


COMMENTS

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.
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
a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal and vertical segments filled in.  Asher Auel, Jan 12 2000
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
Number of right triangles made from vertices of a regular ngon when n is even.  SenPeng Eu, Apr 05 2001
Number of ways to change two nonidentical letters in the word aabbccdd..., where there are n type of letters.  Zerinvary Lajos, Feb 15 2005
a(n) is the number of (n1)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
From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)
Consider a triangle, a pentagon, a heptagon, ..., a kgon where k is odd. We label a triangle with n=1, a pentagon with n=2, ..., a kgon with n = floor(k/2). Imagine a player standing at each vertex of the kgon.
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.
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)
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, Jun 04 2006
If X_1,...,X_n is a partition of a 2nset X into 2blocks then a(n1) is equal to the number of 2subsets of X containing none of X_i, (i=1,...,n).  Milan Janjic, Jul 16 2007
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
Number of (n+1)permutations of 3 objects u,v,w, with repetition allowed, containing n1 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 2permutations either in front, or in the middle, or at the end.  Zerinvary Lajos, Dec 27 2007
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
a(n) is also the least weight of selfconjugate partitions having n different even parts.  Augustine O. Munagi, Dec 18 2008
The general formula for alternating sums of powers of even integers is in terms of the SwissKnife polynomials P(n,x) A153641 (P(n,1)(1)^k P(n,2k+1))/2. Here n=2, thus
a(k) = (P(2,1)  (1)^k*P(2,2k+1))/2. (End)
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
Number of units of a(n) belongs to a periodic sequence: 0, 4, 2, 4, 0.  Mohamed Bouhamida, Sep 04 2009
Number of roots in the root system of type D_{n+1} (for n>2).  Tom Edgar, Nov 05 2013
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
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
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
Consider two equal rectangles composed of unit squares. Then surround the 1st rectangle with 1unitwide 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
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
Also the number of minimum connected dominating sets in the (n+1)cocktail party graph.  Eric W. Weisstein, Jun 29 2017
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 MindBenders", 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
a(n3) is the maximum irregularity over all maximal 2degenerate graphs with n vertices. The extremal graphs are 2stars (K_2 joined to n2 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


REFERENCES

Tom M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 3.
Albert H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: AddisonWesley, 1994.
Peter Winkler, Mathematical MindBenders, Wellesley, Massachusetts: A K Peters, 2007.


LINKS

Eric Weisstein's World of Mathematics, Gear Graph.


FORMULA

a(n) = a(n1)+4*n; o.g.f.:4*x/(1x)^3; e.g.f.: exp(x)*(2*x^2+4*x).  Geoffrey Critzer, May 17 2009
a(n) = 1/int((x*n+x1)*(step((1+x*n)/n)1)*n*step((x*n+x1)/(n+1)),x=0..1) where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((2*n)*(n+1)) = 1/2. (End)
a(n) = 3*a(n1)  3*a(n2) + a(n3); a(0)=0, a(1)=4, a(2)=12.  Harvey P. Dale, Jul 25 2011
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} (sin(x))^(2*n1)*(cos(x))^3).  Francesco Daddi, Aug 02 2011
a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.
t(k)*a(n) + t(k1)*a(n+1) = a((n+1)*(t(k)t(k1)1)), where k>=2, n>=1, t(k)=A000217(k). (End)
Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/2)/(Pi/2).
Product_{n>=1} (1  1/a(n)) = 2*cos(sqrt(3)*Pi/2)/Pi. (End)


EXAMPLE

a(7)=112 because 112 = 2*7*(7+1).
The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...
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


MATHEMATICA

LinearRecurrence[{3, 3, 1}, {0, 4, 12}, 50] (* Harvey P. Dale, Jul 25 2011 *)


PROG

(Haskell)


CROSSREFS

Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.
Cf. similar sequences listed in A299645.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



