%I A000384 M4108 N1705
%S A000384 0,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,
780,
%T A000384 861,946,1035,1128,1225,1326,1431,1540,1653,1770,1891,2016,2145,2278,
%U A000384 2415,2556,2701,2850,3003,3160,3321,3486,3655,3828,4005,4186,4371,4560
%N A000384 Hexagonal numbers: n(2n-1).
%C A000384 Also a(n)=Sum(tan^2((k - 1/2)*pi/(2n)), k, 1, n); - Ignacio Larrosa (ignacio.larrosa(AT)eresmas.net),
Apr 17 2001
%C A000384 Number of edges in the join of two complete graphs, each of order n,
K_n * K_n - Roberto E. Martinez II (remartin(AT)fas.harvard.edu),
Jan 07 2002
%C A000384 The power series expansion of the entropy function H(x) = (1+x)ln(1+x)+(1-x)ln(1-x)
has 1/a_i as coefficient of x^(2i) (the odd terms being zero). -
Tommaso Toffoli (tt(AT)bu.edu), May 06 2002
%C A000384 Partial sums of A016813 (4n+1). Also with offset = 0, a(n) = (2n+1)(n+1)
= A005408 * A000027 = 2n^2 + 3n + 1, i.e. a(0) = 1. - Jeremy Gardiner
(jeremy.gardiner(AT)btinternet.com), Sep 29 2002
%C A000384 Sequence also refers to greatest semiperimeter of primitive Pythagorean
triangles having inradius n-1. Such a triangle has consecutive longer
sides, with short leg 2n-1, hypotenus a(n)-(n-1)=A001844(n) and area
(n-1)*a(n)=6*A000330(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com),
Apr 23 2003
%C A000384 Number of divisors of 12^(n-1), i.e. A000005(A001021(n-1)). - Henry Bottomley
(se16(AT)btinternet.com), Oct 22 2001
%C A000384 Number of standard tableaux of shape (2n-1,1,1) (n>=1). - Emeric Deutsch
(deutsch(AT)duke.poly.edu), May 30 2004
%C A000384 It is well known that for n>0, A014105(n) [0,3,10,21,...] is the first
of 2n+1 consecutive integers such that the sum of the squares of
the first n+1 such integers is equal to the sum of the squares of
the last n; e.g. 10^2+11^2+12^2=13^2+14^2.
%C A000384 Less well known is that for n>1, a(n) [0,1,6,15,28... ] is the first
of 2n consecutive integers such that sum of the squares of the first
n such integers is equal to the sum of the squares of the last n-1
plus n^2; e.g. 15^2+16^2+17^2 = 19^2+20^2+3^2 - Charlie Marion, Dec
16 2006
%C A000384 a(n) is also a perfect number A000396 when n is an even superperfect
number A061652. [From Omar E. Pol (info(AT)polprimos.com), Sep 05
2008]
%C A000384 Sequence arises from reading the line from 0, in the direction 0, 6,...
and the line from 1, in the direction 1, 15,..., in the square spiral
whose vertices are the triangular numbers A000217. [From Omar E.
Pol (info(AT)polprimos.com), Jan 09 2009]
%C A000384 Also, let Hex(n)= hexagonal number, T(n)=triangular number, then Hex(n)=
T(n)+3*T(n-1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Jan 28 2009]
%D A000384 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000384 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000384 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964,
p. 189.
%D A000384 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public.
256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see
vol. 2, p. 2.
%D A000384 Clark Kimberling, Complementary Equations, Journal of Integer Sequences,
Vol. 10 (2007), Article 07.1.4.
%H A000384 T. D. Noe, <a href="b000384.txt">Table of n, a(n) for n=0..1000</a>
%H A000384 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000384 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000384 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000384 Paul Cooijmans, <a href="http://web.archive.org/web/20050302174449/http:/
/members.chello.nl/p.cooijmans/gliaweb/tests/odds.html">Odds</a>.
%H A000384 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=340">
Encyclopedia of Combinatorial Structures 340</a>
%H A000384 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
On Regular Polytope Numbers</a>
%H A000384 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000384 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000384 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
HexagonalNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000384 Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http:/
/www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</
a>, vol. 8 (2008).
%F A000384 E.g.f.: exp(x)(x+2x^2) - Paul Barry (pbarry(AT)wit.ie), Jun 09 2003
%F A000384 G.f.: x(1+3x)/(1-x)^3. a(n)=A000217(2n-1)=A014105(-n).
%F A000384 a(n)=4*A000217(n-1) + n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun
03 2004
%F A000384 a(n) = right term of M^n * [1,0,0], where M = the 3 X 3 matrix [1,0,0;
1,1,0; 1,4,1]. Example: a(5) = 45 since M^5 *[1,0,0] = [1,5,45].
- Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2006
%F A000384 Row sums of triangle A131914. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jul 27 2007
%F A000384 Row sums of n-th row, triangle A134234 starting (1, 6, 15, 28,...). -
Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 14 2007
%F A000384 Starting with offset 1, = binomial transform of [1, 5, 4, 0, 0, 0,...].
Also, A004736 * [1, 4, 4, 4,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 25 2007
%F A000384 a(n)^2+(a(n)+1)^2+...+(a(n)+n-1)^2=(a(n)+n+1)^2+...(a(n)+2n-1)^2+n^2;
e.g., 6^2+7^2=9^2+2^2; 28^+29^2+30^2+31^2=33^2+34^2+35^2+4^2 - Charlie
Marion (charliemath(AT)optonline.net), Nov 10 2007
%F A000384 a(n) = C(n+1,2) + 3 C(n,2)
%F A000384 a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=6 [From Jaume Oliver
Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
%F A000384 a(n)=4*n+a(n-1)-7 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 08 2009]
%e A000384 For n=2, a(2)=4*2+0-7=1; n=3, a(3)=4*3+1-7=6; n=4, a(4)=4*4+6-7=15 [From
Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A000384 [seq (stirling2(2*n,1)*binomial(2*n,2),n=0..48)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo),
Dec 06 2006
%p A000384 a:=n->sum(n/2, j=2..n): seq(a(2*n), n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 30 2007
%p A000384 A000384:=-(1+3*z)/(z-1)^3; [S. Plouffe in his 1992 dissertation, dropping
the initial zero.]
%p A000384 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+4 od: seq(a[n],
n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18
2008
%p A000384 with(finance):seq(add(cashflows([n,k,k], 0 ),k=0..n-1),n=0..51); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008
%p A000384 a:=n->sum(1+sum(2, k=1..n), k=0..n):seq(a(n), n=0...43); [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
%t A000384 Array[ #*(2*#-1)&,20,0] - Vladimir Orlovsky (4vladimir(AT)gmail.com),
Apr 29 2008
%t A000384 Table[2*n^2 + 3*n + 1, {n, -1, 46}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 10 2009]
%o A000384 (PARI) a(n)=n*(2*n-1)
%o A000384 (Other) sage: [2*n*bernoulli_polynomial(n,1) for n in xrange(0,49)] [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2009]
%Y A000384 Cf. A000217, A014105, A077616.
%Y A000384 a(n)= A093561(n+1, 2), (4, 1)-Pascal column.
%Y A000384 a(n)=A100345(n, n-1) for n>0.
%Y A000384 Cf. A131914.
%Y A000384 Cf. A134235.
%Y A000384 Cf. A004736.
%Y A000384 Cf. A000217, A000326, A000566.
%Y A000384 Cf. A000396, A061652. [From Omar E. Pol (info(AT)polprimos.com), Sep
05 2008]
%Y A000384 Cf. A014634, A014635. [From Omar E. Pol (info(AT)polprimos.com), Jan
09 2009]
%Y A000384 Sequence in context: A094142 A081873 A096892 this_sequence A164000 A134978
A115742
%Y A000384 Adjacent sequences: A000381 A000382 A000383 this_sequence A000385 A000386
A000387
%K A000384 nonn,easy,nice
%O A000384 0,3
%A A000384 N. J. A. Sloane (njas(AT)research.att.com).
%E A000384 More terms from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2006
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