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A046092 4 times triangular numbers: a(n) = 2*n*(n+1). 136
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 (list; graph; refs; listen; history; text; internal format)
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 (asher.auel(AT)reed.edu), 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 n-gon when n is even. - Sen-Peng Eu, Apr 05 2001

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

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

From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)

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 every player standing on every vertex of the k-gon.

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.

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. (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

A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007

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

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

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

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

Twice oblong numbers. - Omar E. Pol, May 03 2008

a(n) is also the least weight of self-conjugate partitions having n different even parts. - Augustine O. Munagi, Dec 18 2008

From Peter Luschny, Jul 12 2009: (Start)

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

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 (bhmd95(AT)yahoo.fr), 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) = A245300(n,n). - Reinhard Zumkeller, Jul 17 2014

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 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

  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

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.

A. 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: Addison-Wesley, 1994.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

H. J. Brothers, Pascal's Prism: Supplementary Material.

Milan Janjic, Two Enumerative Functions

Ron Knott, Pythagorean Triples and Online Calculators

Tanya Khovanova, A Miracle Equation [From Tanya Khovanova, Jul 20 2009]

A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501. [From Augustine O. Munagi, Dec 18 2008]

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.

Eric Weisstein's World of Mathematics, Aztec Diamond

Eric Weisstein's World of Mathematics, Cocktail Party Graph

Eric Weisstein's World of Mathematics, Connected Dominating Set

Eric Weisstein's World of Mathematics, Gear Graph

Eric Weisstein's World of Mathematics, Hamiltonian Path

Eric Weisstein's World of Mathematics, Pythagorean Triple

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = A100345(n+1, n-1) for n>0.

a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy, May 25 2004

a(n) = C(2n, 2) - n = 4*C(n, 2). - Zerinvary Lajos, Feb 15 2005

a(n)=A028896-A002378; a(n)=A124080-A028896; a(n)=A049598-A033996. - Zerinvary Lajos, Mar 06 2007

Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol, May 03 2008

a(n) = a(n-1)+4*n; o.g.f.:4*x/(1-x)^3; e.g.f.: exp(x)*(2*x^2+4*x). - Geoffrey Critzer, May 17 2009

From Stephen Crowley, Jul 26 2009: (Start)

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

sum(1/a(n), n >= 1) = sum(1/((2*n)*(n+1)), n >= 1) = 1/2. (End)

a(0)=0, a(1)=4, a(2)=12, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Jul 25 2011

For n>0 a(n)=1/(Integral_{x=0..Pi/2} (sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011

a(n) = A001844(n) - 1. - Omar E. Pol, Oct 03 2011

(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

From Ivan N. Ianakiev, Aug 30 2013: (Start)

a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.

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)

2*a(n)+1 = A016754(n) = A005408(n)^2, the odd squares. - M. F. Hasler, Oct 02 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 = A187832. - Ilya Gutkovskiy, Mar 16 2017

a(n) = lcm(2*n,2*n+2). - Enrique Navarrete, Aug 30 2017

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

Table[2 n (n + 1), {n, 0, 50}] (* Stefan Steinerberger, Apr 03 2006 *)

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

4*Binomial[Range[50], 2] (* Harvey P. Dale, Jul 25 2011 *)

PROG

(PARI) a(n)=binomial(n+1, 2)<<2 \\ Charles R Greathouse IV, Jun 10 2011

(MAGMA) [2*n*(n+1): n in [0..50]]; // Vincenzo Librandi, Oct 04 2011

(Maxima) A046092(n):=2*n*(n+1)$

makelist(A046092(n), n, 0, 30); /* Martin Ettl, Nov 08 2012 */

(Haskell)

a046092 = (* 2) . a002378  -- Reinhard Zumkeller, Dec 15 2013

CROSSREFS

Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.

Main diagonal of array in A001477.

Equals A033996/2 Cf. A001844. - Augustine O. Munagi, Dec 18 2008

Cf. A078371, A141530 (see Librandi's comment in A078371).

Cf. A097080, A001845.

Sequence in context: A008006 A081937 A088557 * A008241 A008216 A008074

Adjacent sequences:  A046089 A046090 A046091 * A046093 A046094 A046095

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Eric W. Weisstein

STATUS

approved

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Last modified October 23 14:42 EDT 2017. Contains 293789 sequences.