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A000290 The squares: a(n) = n^2.
(Formerly M3356 N1350)
1305
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, 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 06 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

Numbers a such that a^1/2 + b^1/2 = c^1/2 and a^2 + b = c. - Cino Hilliard, Feb 07 2008 (this comment needs clarification, Joerg Arndt, Sep 12 2013)

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). - Gary W. Adamson, Aug 17 2008

Equals row sums of triangles A143595 and A056944. - 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. - Paul Curtz, Nov 06 2008

a(n) is the number of all partitions of the sum 2^2+2^2+...2^2, (n-1)-times, into powers of 2. - 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. - Srikanth K S, Jun 25 2009

Zero together with the numbers n such that 2=number of perfect partitions of n. - Juri-Stepan Gerasimov, Sep 26 2009

Totally multiplicative sequence with a(p) = p^2 for prime p. - Jaroslav Krizek, Nov 01 2009

Satisfies A(x)/A(x^2), A(x) = A173277: (1, 4, 13, 32, 74,...). - Gary W. Adamson, Feb 14 2010

a(n) = 1 (mod n+1). - 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. - 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

Drmota, Mauduit, and Rivat proved that the Thue-Morse sequence along the squares is normal; see A228039. - Jonathan Sondow, Sep 03 2013

a(n) can be decomposed into the sum of the four numbers [binomial(n,1)+binomial(n,2)+binomial(n-1,1)+binomial(n-2,2)] which form a "square" in Pascal's Triangle A007318, or the sum of the two numbers [binomial(n,2)+binomial(n+1,2)], or the difference of the two numbers [binomial(n+2,3)-(binomial(n,3)]. - John Molokach, Sep 26 2013

In terms of triangular tiling the number of equilateral triangles with side length 1 inside an equilateral triangle with side length n. - _Kai G. Stier_, Oct 30 2013

Number of positive roots in the root systems of type B_n and C_n (when n>1). - Tom Edgar, Nov 05 2013

Squares of squares (fourth powers) are also called biquadratic numbers: A000290 o A000290 = A000583. - M. F. Hasler, Dec 29 2013

REFERENCES

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.

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.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

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).

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.

LINKS

Franklin T. Adams-Watters, The first 10000 squares: Table of n, n^2 for n = 0..10000

Valentin P. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 10-12.

H. Bottomley, Some Smarandache-type multiplicative sequences

J. Derbyshire, Monkeys and Doors

M. Drmota, C. Mauduit, J. Rivat, The Thue-Morse Sequence Along The Squares is Normal, Abstract, ÖMG-DMV Congress, 2013.

Ralph Greenberg, Math for Poets

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 338

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Milan Janjic, Two Enumerative Functions

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

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, 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).

FORMULA

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. - 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. - Jaume Oliver Lafont, Nov 18 2008

a(n) = floor [ n*(n+1)* [sum_{i=1..n} 1/(n*(n+1))]]. - Ctibor O. Zizka, Mar 07 2009

Product_{i=2..infinity} (1-2/a(i)) = -sin(A063448)/A063448. - 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. - David Scheers (dscheers(AT)webpoint.nl), Mar 15 2009

a(n) = A002378(n-1) + n. - 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). - Bruno Berselli, May 04 2010

a(n) = a(n-1)+a(n-2)-a(n-3)+4, n>2. - 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 polynomial of order n and Qn the secondary Laguerre polynomial defined by Qn(x) = integral{t=0..infinity} (Pn(x)-Pn(t))*exp(-t)/(x-t). - 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

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

a(n) = A133280(A000217(n)). - Ivan N. Ianakiev, Aug 13 2013

a(2*a(n)+2*n+1) = a(2*a(n)+2*n) + a(2*n+1). - Vladimir Shevelev, Jan 24 2014

a(n+1) = sum(t1+2*t2+...+n*tn=n, (-1)^(n+t1+t2+...+tn)*multinomial(t1+t2 +...+tn,t1,t2,...,tn)*4^(t1)*7^(t2)*8^(t3+...+tn). - Mircea Merca, Feb 27 2014

EXAMPLE

Example: A000290=F=25. n=5. Q=10. Q^2=F*4 => 10^2=25*4=100. - 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. - Bruno Berselli, May 04 2010

G.f. = 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 + ...

MAPLE

A000290 := n->n^2; seq(A000290(n), n=0..50);

A000290 := -(1+z)/(z-1)^3; # [Simon Plouffe, in his 1992 dissertation, for sequence starting at a(1).]

MATHEMATICA

a[n_] := n^2; Table[a[n], {n, 0, 50}] (* Stefan Steinerberger, Mar 30 2006*)

Array[#^2 &, 51, 0] (* Robert G. Wilson v, Aug 01 2014 *)

PROG

(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 */

CROSSREFS

Cf. A092205, A128200, A005408, A128201, A002522, A005563, A008865, A059100, A143051, A143470, A143595, A056944, A001788 (binomial transform), A228039, A001105, A004159, A159918, A173277, A095794, A162395, A186646 (Pisano periods).

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. - Valentin Bakoev (v_bakoev(AT)yahoo.com), Mar 03 2009

Cf. n-gonal numbers: A000217, this sequence, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.

Boustrophedon transforms: A000697, A000745.

Sequence in context: A174452 A174902 * A162395 A221222 A144913 A018885

Adjacent sequences:  A000287 A000288 A000289 * A000291 A000292 A000293

KEYWORD

nonn,core,easy,nice,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

Incorrect comment and example removed by Joerg Arndt, Mar 11 2010

Broken link to Hyun Kwang Kim's paper fixed by Felix Fröhlich, Jun 16 2014

STATUS

approved

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Last modified August 27 19:27 EDT 2014. Contains 246147 sequences.