%I A000290 M3356 N1350
%S A000290 0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,
%T A000290 324,361,400,441,484,529,576,625,676,729,784,841,900,961,
%U A000290 1024,1089,1156,1225,1296,1369,1444,1521,1600,1681,1764,1849
%N A000290 The squares: a(n) = n^2.
%C A000290 Zero followed by partial sums of A005408 (odd numbers). - Jeremy Gardiner
(jeremy.gardiner(AT)btinternet.com), Aug 13 2002
%C A000290 Begin with n, add the next number, subtract the previous number and so
on ending with subtracting a 1: a(n) = n + (n+1) - (n-1) +(n+2) -(n-2)
+(n+3)-(n-3)...+(2n-1)-1 = n^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
Mar 24 2004
%C A000290 Sum of two consecutive triangular numbers A000217. - Lekraj Beedassy
(blekraj(AT)yahoo.com), May 14 2004
%C A000290 Numbers with an odd number of divisors: {d(n^2)=A048691(n); for the first
occurrence of 2n+1 divisors, see A071571(n)}. - Lekraj Beedassy (blekraj(AT)yahoo.com),
Jun 30 2004. See also A000037.
%C A000290 First sequence ever computed by electronic computer, on EDSAC, May 6
1949 (see Renwick link). - Russ Cox (rsc(AT)swtch.com), Apr 20 2006
%C A000290 Numbers n such that the imaginary quadratic field Q[Sqrt[ -n]] has four
units. - Marc LeBrun (mlb(AT)well.com), Apr 12 2006
%C A000290 Number of permutations of two distinct letters (AB) or numbers each of
which appears n=0 to infinity ("0", AB, AABB, AAABBB, AAAABBBB, AAAAABBBBB,
etc...)with two and n-2 fixed points. - Zerinvary Lajos (zerinvarylajos@yahoo.com),
Nov 12 2009
%C A000290 For n>0: number of divisors of (n-1)th power of any squarefree semiprime:
a(n)=A000005(A006881(k)^(n-1)); a(n) = A000005(A000400(n-1)) = A000005(A011557(n-1))
= A000005(A001023(n-1)) = A000005(A001024(n-1)). - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Mar 04 2007
%C A000290 For n>=1, a(n) is equal to the number of functions f:{1,2}->{1,2,...,
n} such that for y_1, y_2 in {1,2,...,n} we have f(1)<>y_1 and f(2)<>
y_2. - Milan R. Janjic (agnus(AT)blic.net), Apr 17 2007
%C A000290 If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then
a(n-2) is the number of 3-subsets of X intersecting both Y and Z.
- Milan R. Janjic (agnus(AT)blic.net), Sep 19 2007
%C A000290 Also numbers a such that a^1/2 + b^1/2 = c^1/2 and a^2 + b = c. - Cino
Hilliard (hillcino368(AT)hotmail.com), Feb 07 2008
%C A000290 Numbers n such that the geometric mean of the divisors of n is an integer.
- Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Jun 26 2008
%C A000290 Equals row sums of triangle A143470. Example: 36 = sum of row 6 terms:
(23 + 7 + 3 + 1 + 1 + 1). [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Aug 17 2008]
%C A000290 Equals row sums of triangles A143595 and A056944 [From Gary W. Adamson
(qntmpkt(AT)yahoo.com), Aug 26 2008]
%C A000290 Number of divisors of 6^n - J. Lowell (jhbubby(AT)mindspring.com), Aug
30 2008
%C A000290 Denominators of Lyman spectrum of hydrogen atom. Numerators are A005563.
A000290-A005563=A000012. Just before A061038,A061040,A061042,A061044,
A061046,A061048,A061050. See A035287. [From Paul Curtz (bpcrtz(AT)free.fr),
Nov 06 2008]
%C A000290 Number n such that if a=n, b=2n, c=3n, d=6n*sqrt(n), then a^3+b^3+c^3=d^2
Example: a=1, b=2, c=3, d=6, 1^3+2^3+3^3=6^2; a=16, b=32, c=48, d=384,
16^3+32^3+48^3=384^2 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Jan 23 2009]
%C A000290 a(n) is also the number of all partitions of the sum 2^2+2^2+...2^2,
(n-1)-times, into powers of 2. [From Valentin Bakoev (v_bakoev(AT)yahoo.com),
Mar 03 2009]
%C A000290 a(n) is the maximal number of squares that can be 'on' in an n X n board
so that all the squares turn 'off' after applying the operation :
in any 2 X 2 sub-board, a square turns from 'on' to 'off' if the
other three are off. [From Srikanth K S (sriperso(AT)gmail.com),
Jun 25 2009]
%C A000290 For n>0, if a=n, b=2n, c=3*n*sqrt(n), then a^3+b^3=c^2 [From Vincenzo
Librandi (vincenzo.librandi(AT)tin.it), Jul 02 2009]
%C A000290 Zero together with the numbers n such that 2=number of perfect partitions
of n [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Sep 26
2009]
%C A000290 Totally multiplicative sequence with a(p) = p^2 for prime p. [From Jaroslav
Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 2009]
%D A000290 G. L. Alexanderson et al., The William Lowell Putnam Mathematical Competition,
Problems and Solutions:1965-1984, "December 1967 Problem B4(a)",
pp. 8(157) MAA Washington DC 1985.
%D A000290 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 2.
%D A000290 Bakoev V., Algorithmic approach to counting of certain types m-ary partitions,
Discrete Mathematics, 275 (2004) pp.17-41. [From Valentin Bakoev
(v_bakoev(AT)yahoo.com), Mar 03 2009]
%D A000290 R. P. Burn & A. Chetwynd, A Cascade Of Numbers, "The prison door problem"
Problem 4 pp. 5-7;79-80 Arnold London 1996.
%D A000290 M. Gardner, Time Travel and Other Mathematical Bewilderments, Chapter
6 pp. 71-2, W.H.Freeman NY 1988.
%D A000290 Clark Kimberling, Complementary Equations, Journal of Integer Sequences,
Vol. 10 (2007), Article 07.1.4.
%D A000290 A. S. Posamentier, The Art of Problem Solving, Section 2.4 "The Long
Cell Block" pp. 10-1;12;156-7 Corwin Press Thousand Oaks CA 1996.
%D A000290 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000290 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000290 J. K. Strayer, Elementary Number Theory, Exercise Set 3.3 Problems 32;
33 pp. 88 PWS Publishing Co. Boston MA 1996.
%D A000290 C. W. Trigg, Mathematical Quickies, "The Lucky Prisoners" Problem 141
pp. 40;141 Dover NY 1985.
%D A000290 R. Vakil, A Mathematical Mosaic, "The Painted Lockers" pp. 127;134 Brendan
Kelly Burlington Ontario 1996.
%H A000290 Franklin T. Adams-Watters, <a href="b000290.txt">The first 10000 squares:
Table of n, n^2 for n = 0..10000</a>
%H A000290 V. Bakoev, <a href="http://www.uni-vt.bg/showfullpub.asp?u=32&fn=AACCTMP.pdf">
Algorithmic approach to counting of certain types m-ary partitions</
a>, Discrete Mathematics, 275 (2004) pp. 17-41.
%H A000290 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">
Some Smarandache-type multiplicative sequences</a>
%H A000290 J. Derbyshire, <a href="http://olimu.com/Notes/Monkeys&Doors.htm">Monkeys
and Doors</a>
%H A000290 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=338">
Encyclopedia of Combinatorial Structures 338</a>
%H A000290 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas
for Some Functions on Finite Sets</a>
%H A000290 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000290 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
On Regular Polytope Numbers</a>
%H A000290 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 A000290 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 A000290 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Arithmetic and growth of periodic orbits</a>, J. Integer
Seqs., Vol. 4 (2001), #01.2.1.
%H A000290 W. S. Renwick, <a href="http://www.cl.cam.ac.uk/Relics/elog.html">EDSAC
log</a>.
%H A000290 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol6710.html">
28th Putnam 1967 Prob.B4(a)</a>
%H A000290 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">
Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal
of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H A000290 N. J. A. Sloane, <a href="a217.gif">Illustration of initial terms of
A000217, A000290, A000326</a>
%H A000290 D. Surendran, <a href="http://www.uz.ac.zw/science/maths/zimaths/jailer.htm">
Chimbumu and Chickwama get out of jail</a>
%H A000290 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SquareNumber.html">Link to a section of The World of Mathematics
(1).</a>
%H A000290 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BiquadraticNumber.html">Link to a section of The World of Mathematics
(2).</a>
%H A000290 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Unit.html">Unit</a>
%H A000290 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
WienerIndex.html">Wiener Index</a>
%H A000290 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000290 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000290 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000290 Multiplicative with a(p^e) = p^(2e). - David W. Wilson (davidwwilson(AT)comcast.net),
Aug 01, 2001.
%F A000290 G.f.: x(1+x)/(1-x)^3. E.g.f.: exp(x)(x+x^2). Dirichlet g.f.: zeta(s-2).
a(n)=a(-n).
%F A000290 Sum of all matrix elements M(i, j) = 2*i/(i+j) (i, j = 1..n). a(n) =
Sum[Sum[2*i/(i+j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com),
Oct 24 2004
%F A000290 a(0)=0, a(1)=1, a(n)=2*a(n-1)-a(n-2)+2 - Miklos Kristof (kristmikl(AT)freemail.hu),
Mar 09 2005
%F A000290 a(n)=sum of the odd numbers for i=1 to n. a(0)=0 a(1)=1 then a(n)=a(n-1)+2*n-1.
- Pierre CAMI (pierrecami(AT)tele2.fr), Oct 22 2006
%F A000290 For n>0: a(n) = A130064(n)*A130065(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
May 05 2007
%F A000290 a(n) = Sum(A002024(n,k): 1<=k<=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 24 2007
%F A000290 Left edge of the triangle in A132111: a(n)=A132111(n,0). - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Aug 10 2007
%F A000290 a(n) = {least common multiple of n and n-1} - (n-1). - Mats Granvik (mgranvik(AT)abo.fi),
Sep 16 2007
%F A000290 Binomial transform of [1, 3, 2, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 21 2007
%F A000290 a(n) = C(n+1,2) + C(n,2)
%F A000290 This sequence could be derived from the following general formula (cf.
A001286, A000330): n*(n+1)*...*(n+k)*[n+(n+1)+...+(n+k)]/((k+2)!*(k+1)/
2 ) at k=0 Indeed, using the formula for the sum of the arithmetic
progression [n+(n+1)+...+(n+k)]= (2*n + k)*(k + 1)/2 the general
formula could be rewritten as: n*(n+1)*...*(n+k)*(2*n + k)/(k+2)!
so for k=0 above general formula degenerates to n*(2*n + 0)/(0+2)!=
n^2 - Alexander R. Povolotsky (pevnev(AT)juno.com), May 18 2008
%F A000290 From a(4) recurence formula a(n+3)=3a(n+2)-3a(n+1)+a(n) and a(1)=1, a(2)=4,
a(3)=9 [From Artur Jasinski (grafix(AT)csl.pl), Oct 21 2008]
%F A000290 The recurrence a(n+3)=3a(n+2)-3a(n+1)+a(n) is satisfied by all k-gonal
sequences from a(3), with a(0)=0, a(1)=1, a(2)=k. [From Jaume Oliver
Lafont (joliverlafont(AT)gmail.com), Nov 18 2008]
%F A000290 a(n) = floor [ n*(n+1)* [sum_{i=1..n} 1/(n*(n+1))]] [From Ctibor O. Zizka
(c.zizka(AT)email.cz), Mar 07 2009]
%F A000290 Product_{i=2..infinity} (1-2/a(i)) = -sin(A063448)/A063448. [From R.
J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 2009]
%F A000290 let this A000290=F(actor) then F*4=Q^2 always where Q=2*n if n>=0 and
n are the unique numbers of exact roots Q. [From david scheers (dscheers(AT)webpoint.nl),
Mar 15 2009]
%F A000290 G.f.: x*(1-x)/exp(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 03 2009]
%F A000290 a(n) = A002378(n-1) + n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz),
Jun 14 2009]
%e A000290 example: A000290=F=25. n=5. Q=10. Q^2=F*4 => 10^2=25*4=100 [From david
scheers (dscheers(AT)webpoint.nl), Mar 15 2009]
%e A000290 For n=1, 1^3+2^3=3^2; n=4, 4^3+8^3=24^2; n=9, 0^3+18^3=81^2 [From Vincenzo
Librandi (vincenzo.librandi(AT)tin.it), Jul 02 2009]
%p A000290 A000290 := n->n^2;
%p A000290 A000290:=-(1+z)/(z-1)^3; [S. Plouffe, in his 1992 dissertation, for sequence
starting at a(1).]
%p A000290 a:=n->sum(1+sum(1, k=3..n),k=2..n):seq(a(n), n=1...44); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 11 2008
%p A000290 with(finance):seq(add(cashflows([k, k, -1], 0 ), k=1..n), n=0..45); #
[From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 22 2008]
%p A000290 restart: G(x):=x*(1-x)/exp(x): f[0]:=G(x): for n from 1 to 43 do f[n]:=diff(f[n-1],
x) od: x:=0: seq(abs(f[n]),n=0..43);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 03 2009]
%t A000290 a[n_] := n^2; Table[a[n], {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com),
Mar 30 2006
%t A000290 Array[ #^2 &, 60, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec
26 2006
%t A000290 a = {1, 4, 9}; k = 1; m = 4; r = 9; Do[l = 3 r - 3 m + k; AppendTo[a,
l]; k = m; m = r; r = l, {n, 1, 100}]; a [From Artur Jasinski (grafix(AT)csl.pl),
Oct 21 2008]
%t A000290 s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,1,5!,2}];lst [From Vladimir Orlovsky
(4vladimir(AT)gmail.com), Apr 02 2009]
%o A000290 (MAGMA) [ n^2 : n in [0..1000]];
%o A000290 (PARI) a(n)=n^2
%o A000290 (Other) sage: [log(e^(n^2))for n in xrange(0,34)]# [From Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 03 2009]
%Y A000290 Cf. A092205, A128200, A005408, A128201, A002522, A005563, A008865, A059100,
A143051, A143470, A143595, A056944.
%Y A000290 A row or column of A132191.
%Y A000290 A000290 is related to partitions of 2^n into powers of 2, as it is shown
in A002577. So A002577 connects A000290 and A000447. [From Valentin
Bakoev (v_bakoev(AT)yahoo.com), Mar 03 2009]
%Y A000290 Cf. A001105 [From David Scheers (dscheers(AT)webpoint.nl), Mar 15 2009]
%Y A000290 A004159, A159918. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Apr 25 2009]
%Y A000290 Sequence in context: A106545 A093837 A069821 this_sequence A162395 A144913
A018885
%Y A000290 Adjacent sequences: A000287 A000288 A000289 this_sequence A000291 A000292
A000293
%K A000290 nonn,core,easy,nice,mult,new
%O A000290 0,3
%A A000290 N. J. A. Sloane (njas(AT)research.att.com).
|
