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A000290
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The squares: a(n) = n^2.
(Formerly M3356 N1350)
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1026
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0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500
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OFFSET
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0,3
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COMMENTS
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To test if a number is a square, see Cohen, p. 40. - N. J. A. Sloane, Jun 19 2011.
Zero followed by partial sums of A005408 (odd numbers). - Jeremy Gardiner, Aug 13 2002
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
Sum of two consecutive triangular numbers A000217. - Lekraj Beedassy, May 14 2004
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, Jun 30 2004. See also A000037.
First sequence ever computed by electronic computer, on EDSAC, May 6 1949 (see Renwick link). - Russ Cox, Apr 20 2006
Numbers n such that the imaginary quadratic field Q[Sqrt[ -n]] has four units. - Marc LeBrun, Apr 12 2006
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, Mar 04 2007
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 Janjic, Apr 17 2007
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 Janjic, Sep 19 2007
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
Numbers n such that the geometric mean of the divisors of n is an integer. - Ctibor O. Zizka, Jun 26 2008
Equals row sums of triangle A143470. Example: 36 = sum of row 6 terms: (23 + 7 + 3 + 1 + 1 + 1). [From Gary W. Adamson , Aug 17 2008]
Equals row sums of triangles A143595 and A056944 [From Gary W. Adamson, Aug 26 2008]
Number of divisors of 6^(n-1) for n>0. - J. Lowell, Aug 30 2008
Denominators of Lyman spectrum of hydrogen atom. Numerators are A005563. A000290-A005563=A000012. [From Paul Curtz, Nov 06 2008]
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]
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, Jun 25 2009]
Zero together with the numbers n such that 2=number of perfect partitions of n [From Juri-Stepan Gerasimov, Sep 26 2009]
Totally multiplicative sequence with a(p) = p^2 for prime p. [From Jaroslav Krizek, Nov 01 2009]
Satisfies A(x)/A(x^2), A(x) = A173277: (1, 4, 13, 32, 74,...) [From Gary W. Adamson, Feb 14 2010]
a(n) = 1 (mod n+1). [From Bruno Berselli, Jun 03 2010]
Positive members are the integers with an odd number of odd divisors and an even number of even divisors. See also A120349, A120359, A181792, A181793, A181795. [From Matthew Vandermast, Nov 14 2010]
A007968(a(n)) = 0. [Reinhard Zumkeller, Jun 18 2011]
A071974(a(n)) = n; A071975(a(n)) = 1. [Reinhard Zumkeller, Jul 10 2011]
Besides the first term this sequence is the denominator of ((pi)^2)/6=1+1/4+1/9+1/16+1/25+1/36+... - Mohammad K. Azarian, Nov 01 2011
Partial sums give A000330. - Omar E. Pol, Jan 12 2013
a(n) = A162610(n,n) = A209297(n,n) for n > 0. - Reinhard Zumkeller, Jan 19 2013
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REFERENCES
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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.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
Bakoev V., Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp.17-41.
R. P. Burn & A. Chetwynd, A Cascade Of Numbers, "The prison door problem" Problem 4 pp. 5-7;79-80 Arnold London 1996.
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, page 40.
M. Gardner, Time Travel and Other Mathematical Bewilderments, Chapter 6 pp. 71-2, W.H.Freeman NY 1988.
L. B. W. Jolley, Summation of Series, Dover (1961)
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. K. Strayer, Elementary Number Theory, Exercise Set 3.3 Problems 32;33 pp. 88 PWS Publishing Co. Boston MA 1996.
C. W. Trigg, Mathematical Quickies, "The Lucky Prisoners" Problem 141 pp. 40;141 Dover NY 1985.
R. Vakil, A Mathematical Mosaic, "The Painted Lockers" pp. 127;134 Brendan Kelly Burlington Ontario 1996.
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LINKS
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Franklin T. Adams-Watters, The first 10000 squares: Table of n, n^2 for n = 0..10000
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.
H. Bottomley, Some Smarandache-type multiplicative sequences
J. Derbyshire, Monkeys and Doors
Ralph Greenberg, Math for Poets
Guo-Niu Han, Enumeration of Standard Puzzles
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 338
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic, Two Enumerative Functions
Hyun Kwang Kim, On Regular Polytope Numbers
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
W. S. Renwick, EDSAC log.
J. Scholes, 28th Putnam 1967 Prob.B4(a)
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326
M. Somos, Rational Function Multiplicative Coefficients
D. Surendran, Chimbumu and Chickwama get out of jail
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Biquadratic Number
Eric Weisstein's World of Mathematics, Unit
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for "core" sequences
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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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).
Multiplicative with a(p^e) = p^(2e). - David W. Wilson, Aug 01, 2001.
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, Oct 24 2004
a(0)=0, a(1)=1, a(n)=2*a(n-1)-a(n-2)+2 - Miklos Kristof, Mar 09 2005
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, Oct 22 2006
For n>0: a(n) = A130064(n)*A130065(n). - Reinhard Zumkeller, May 05 2007
a(n) = Sum(A002024(n,k): 1<=k<=n). - Reinhard Zumkeller, Jun 24 2007
Left edge of the triangle in A132111: a(n)=A132111(n,0). - Reinhard Zumkeller, Aug 10 2007
Binomial transform of [1, 3, 2, 0, 0, 0,...]. - Gary W. Adamson, Nov 21 2007
a(n) = binomial(n+1,2) + binomial(n,2).
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, May 18 2008
From a(4) recurrence 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, Oct 21 2008]
The recurrence a(n+3)=3*a(n+2)-3*a(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, Nov 18 2008]
a(n) = floor [ n*(n+1)* [sum_{i=1..n} 1/(n*(n+1))]] [From Ctibor O. Zizka, Mar 07 2009]
Product_{i=2..infinity} (1-2/a(i)) = -sin(A063448)/A063448. [From R. J. Mathar, Mar 12 2009]
Let 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]
a(n) = A002378(n-1) + n. [From Jaroslav Krizek, Jun 14 2009]
a(n) = n*A005408(n-1) - sum [i = 1 ... n-2] A005408(i) - (n-1) = n*A005408(n-1) - a(n-1) - (n-1) [From Bruno Berselli, May 04 2010]
a(n) = a(n-1)+a(n-2)-a(n-3)+4, n>2 [from Gary Detlefs, Sep 07 2010]
a(n+1) = integral{x=0..infinity} exp(-x)/( (Pn(x)*exp(-x)*Ei(x)-Qn(x))^2 +(Pi*exp(-x)*Pn(x))^2 ), with Pn the Laguerre polynom of order n and Qn the secondary Laguerre polynom defined by Qn(x) = integral{t=0..infinity} (Pn(x)-Pn(t))*exp(-t)/(x-t) [From Groux Roland, Dec 08 2010]
Euler transform of length 2 sequence [ 4, -1]. - Michael Somos Feb 12 2011
A162395(n) = -(-1)^n * a(n). - Michael Somos Mar 19 2011
a(n) = A004201(A000217(n)); A007606(a(n)) = A000384(n); A007607(a(n)) = A001105(n). [Reinhard Zumkeller, Feb 12 2011]
Sum_{n>=1} 1/a(n)^k = (2*Pi)^k*B_k/(2*k!)=zeta(2k) with Bernoulli numbers B_k = -1, 1/6, 1/30, 1/42,.. for k>=0. See A019673, A195055/10 etc [Jolley eq 319].
Sum_{n>=1} (-1)^(n+1)/a(n)^k = 2^(k-1)*Pi^k*(1-1/2^(k-1))*B_k/k! [Jolley eq 320] with B_k as above.
a(n) = A199332(2*n-1,n). [Reinhard Zumkeller, Nov 23 2011]
For n>1, a(n) = sum(d|n, phi(d)*psi(d)), where phi is A000010 and psi is A001615. - Enrique Pérez Herrero, Feb 29 2012
a(n) = A000217(n^2) - A000217(n^2-1), for n>0. [Ivan N. Ianakiev, May 30 2012]
G.f.: x*(U(0) where U(k)= 1 + 4*x*(k+1)^2/( (2*k+1)^2 - x*(2*k+1)^2*(2*k+3)^2/(x*(2*k+3)^2 +4*(k+1)^2/U(k+1) ));(continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Dec 03 2012
a(n) = (A000217(n) + A000326(n))/2. - Omar E. Pol, Jan 11 2013
a(A000217(n)) = sum(i=1..n, sum(j=1..n, ij)), for n>0. - Ivan N. Ianakiev, Apr 20 2013
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EXAMPLE
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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]
For n = 8, a(8) = 8*15 - (1+3+5+7+9+11+13) - 7 = 8*15 - 49 - 7 = 64 [From Bruno Berselli, May 04 2010]
x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 + 49*x^7 + 64*x^8 + 81*x^9 + ...
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MAPLE
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A000290 := n->n^2;
A000290:=-(1+z)/(z-1)^3; [Simon Plouffe, in his 1992 dissertation, for sequence starting at a(1).]
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MATHEMATICA
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a[n_] := n^2; Table[a[n], {n, 0, 50}] - Stefan Steinerberger, Mar 30 2006
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PROG
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(MAGMA) [ n^2 : n in [0..1000]];
(PARI) {a(n) = n^2}
(Haskell)
a000290 = (^ 2)
a000290_list = scanl (+) 0 [1, 3..] -- Reinhard Zumkeller, Apr 06 2012
(Maxima) A000290(n):=n^2$ makelist(A000290(n), n, 0, 30); /*Martin Ettl, Oct 25 2012*/
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CROSSREFS
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Cf. A092205, A128200, A005408, A128201, A002522, A005563, A008865, A059100, A143051, A143470, A143595, A056944, A001788 (binomial transform).
A row or column of A132191.
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]
Cf. A001105, A004159, A159918, A173277, A095794, A162395, A186646 (Pisano periods)
Cf. n-gonal numbers: A000217, this sequence, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.
Sequence in context: A174452 A174902 * A162395 A221222 A144913 A018885
Adjacent sequences: A000287 A000288 A000289 * A000291 A000292 A000293
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KEYWORD
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nonn,core,easy,nice,mult
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Removed incorrect comment and example Joerg Arndt, Mar 11 2010
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STATUS
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approved
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