

A000290


The squares: a(n) = n^2.
(Formerly M3356 N1350)


2475



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

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)  (n1) + (n+2)  (n2) + (n+3)  (n3) + ... + (2n1)  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 (n1)th power of any squarefree semiprime: a(n) = A000005(A006881(k)^(n1)); a(n) = A000005(A000400(n1)) = A000005(A011557(n1)) = A000005(A001023(n1)) = A000005(A001024(n1)).  Reinhard Zumkeller, Mar 04 2007
If a 2set Y and an (n2)set Z are disjoint subsets of an nset X then a(n2) is the number of 3subsets 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^(n1) 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, (n1)times, into powers of 2.  Valentin Bakoev, 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 subboard, 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 is the number of perfect partitions of n.  JuriStepan 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
Drmota, Mauduit, and Rivat proved that the ThueMorse 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(n1, 1) + binomial(n2, 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.  K. 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: A000583.  M. F. Hasler, Dec 29 2013
For n > 0, a(n) is the largest integer k such that k^2 + n is a multiple of k + n. More generally, for m > 0 and n > 0, the largest integer k such that k^(2*m) + n is a multiple of k + n is given by k = n^(2*m).  Derek Orr, Sep 03 2014
For n > 0, a(n) is the number of compositions of n + 5 into n parts avoiding the part 2.  Milan Janjic, Jan 07 2016
a(n), for n >= 3, is also the number of all connected subtrees of a cycle graph, having n vertices.  Viktar Karatchenia, Mar 02 2016
On every sequence of natural continuous numbers with an even number of elements, the summatory of the second half of the sequence minus the summatory of the first half of the sequence is always a square. Example: Sequence from 61 to 70 has an even number of elements (10). Then 61 + 62 + 63 + 64 + 65 = 315;66 + 67 + 68 + 69 + 70 = 340;340  315 = 25. (n/2)^2 for n = number of elements.  César Aguilera, Jun 20 2016
On every sequence of natural continuous numbers from n^2 to (n+1)^2, the sum of the differences of pairs of elements of the two halves in every combination possible is always (n+1)^2.  César Aguilera, Jun 24 2016
Suppose two circles with radius 1 are tangent to each other as well as to a line not passing through the point of tangency. Create a third circle tangent to both circles as well as the line. If this process is continued, a(n) for n > 0 is the reciprocals of the radii of the circles, beginning with the largest circle.  Melvin Peralta, Aug 18 2016
Does not satisfy Benford's law [Ross, 2012]  N. J. A. Sloane, Feb 08 2017
Numerators of the solution to the generalization of the Feynman triangle problem, with an offset of 2. If each vertex of a triangle is joined to the point (1/p) along the opposite side (measured say clockwise), then the area of the inner triangle formed by these lines is equal to (p  2)^2/(p^2  p + 1) times the area of the original triangle, p > 2. For example, when p = 3, the ratio of the areas is 1/7. The denominators of the ratio of the areas is given by A002061. [Cook & Wood, 2004]  Joe Marasco, Feb 20 2017
Equals row sums of triangle A004737, n >= 1.  Martin Michael Musatov, Nov 07 2017


REFERENCES

G. L. Alexanderson et al., The William Lowell Putnam Mathematical Competition, Problems and Solutions: 19651984, "December 1967 Problem B4(a)", pp. 8(157) MAA Washington DC 1985.
T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 2.
R. P. Burn & A. Chetwynd, A Cascade Of Numbers, "The prison door problem" Problem 4 pp. 57; 7980 Arnold London 1996.
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, p. 40.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), p. 6.
M. Gardner, Time Travel and Other Mathematical Bewilderments, Chapter 6 pp. 712, 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, McGrawHill Book Company, New York (1968), p. 982.
Alfred S. Posamentier, The Art of Problem Solving, Section 2.4 "The Long Cell Block" pp. 101; 12; 1567 Corwin Press Thousand Oaks CA 1996.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence  see "List of Sequences" in Vol. 2.
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, p. 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. AdamsWatters, The first 10000 squares: Table of n, n^2 for n = 0..10000
Valentin P. Bakoev, Algorithmic approach to counting of certain types mary partitions, Discrete Mathematics, 275 (2004) pp. 1741.
Stefano Barbero, Umberto Cerruti, Nadir Murru, Transforming Recurrent Sequences by Using the Binomial and Invert Operators, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 1012.
Henry Bottomley, Some Smarandachetype multiplicative sequences
R. J. Cook and G. V. Wood, Feynman's Triangle, Mathematical Gazette, 88:299302 (2004).
John Derbyshire, Monkeys and Doors
Michael Drmota, Christian Mauduit, Joël Rivat, The ThueMorse Sequence Along The Squares is Normal, Abstract, ÖMGDMV Congress, 2013.
Ralph Greenberg, Math for Poets
GuoNiu Han, Enumeration of Standard Puzzles
GuoNiu Han, Enumeration of Standard Puzzles [Cached copy]
Vi Hart, Doodling in Math Class: Connecting Dots (2012) [Video]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 338
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265282.
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), 6575.
J. H. McKay, The William Lowell Putnam Mathematical Competition, Problem B4(a), The American Mathematical Monthly, vol. 75, no. 7, 1968, pp. 732739.
Matthew Parker, The first million squares (7Zip compressed file)
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
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
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
William S. Renwick, EDSAC log.
Luis Manuel Rivera, Integer sequences and kcommuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 3642.
John 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
Michael Somos, Rational Function Multiplicative Coefficients
Dinoj 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 linear recurrences with constant coefficients, signature (3,3,1).
Index entries for twoway infinite sequences
Index to sequences related to polygonal numbers
Index entries for sequences related to Benford's law


FORMULA

G.f.: x*(1 + x) / (1  x)^3.
E.g.f.: exp(x)*(x + x^2).
Dirichlet g.f.: zeta(s2).
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_{i = 1..n}{j = 1..n} 2*i/(i + j).  Alexander Adamchuk, Oct 24 2004
a(0) = 0, a(1) = 1, a(n) = 2*a(n1)  a(n2) + 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(n1) + 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_{k = 1..n} A002024(n, k).  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) = 3*a(n+2)  3*a(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 kgonal 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} 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, Mar 15 2009
a(n) = A002378(n1) + n.  Jaroslav Krizek, Jun 14 2009
a(n) = n*A005408(n1)  [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} 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} (Pn(x)  Pn(t))*exp(t)/(xt).  Groux Roland, Dec 08 2010
Euler transform of length2 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^(k1)*Pi^k*(11/2^(k1))*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_{dn} 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(n) = A162610(n, n) = A209297(n, n) for n > 0.  Reinhard Zumkeller, Jan 19 2013
a(A000217(n)) = Sum_{i = 1..n}{j = 1..n} i*j, 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
a(n) = floor(1/(1cos(1/n)))/2 = floor(1/(1n*sin(1/n)))/6, n > 0.  Clark Kimberling, Oct 08 2014
a(n) = ceiling(Sum_{k >= 1} log(k)/k^(1+1/n)) = Zeta'[1+1/n]. Thus any exponent greater than 1 applied to k yields convergence. The fractional portion declines from A073002 = 0.93754... at n = 1 and converges slowly to 0.9271841545163232... for large n.  Richard R. Forberg, Dec 24 2014
a(n) = Sum_{j = 1..n}{i = 1..n} ceiling((i + j  n + 1)/3).  Wesley Ivan Hurt, Mar 12 2015
a(n) = Prod_{j = 1..n  1} 2  2*cos(2*j*Pi/n).  Michel Marcus, Jul 24 2015
From Ilya Gutkovskiy, Jun 21 2016: (Start)
Product_{n >= 1} (1 + 1/a(n)) = sinh(Pi)/Pi = A156648.
Sum_{n >= 0} 1/a(n!) = BesselI(0, 2) = A070910. (End)
a(n) = A028338(n, n1), n >= 1 (second diagonal).  Wolfdieter Lang, Jul 21 2017


EXAMPLE

Example: A000290 = F = 25. n = 5. Q = 10. Q^2 = F * 4 => 10^2 = 25 * 4 = 100.  David Scheers, 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 + ...
a(4) = 16. For n = 4 vertices, the cycle graph C4 is ABCDA. The subtrees are: 4 singles: A, B, C, D; 4 pairs: AB, BC, CD, AD; 4 triples: ABC, BCD, CDA, DAB; 4 quads: ABCD, BCDA, CDAB, DABC; 4 + 4 + 4 + 4 = 16.  Viktar Karatchenia, Mar 02 2016


MAPLE

A000290 := n>n^2; seq(A000290(n), n=0..50);
A000290 := (1+z)/(z1)^3; # Simon Plouffe, in his 1992 dissertation, for sequence starting at a(1)


MATHEMATICA

Array[#^2 &, 51, 0] (* Robert G. Wilson v, Aug 01 2014 *)
LinearRecurrence[{3, 3, 1}, {0, 1, 4}, 60] (* Vincenzo Librandi, Jul 24 2015 *)
CoefficientList[Series[(x^2 + x)/(x  1)^3, {x, 0, 50}], x] (* Robert G. Wilson v, Jul 23 2018 *)
Range[0, 99]^2 (* Alonso del Arte, Nov 21 2019 *)


PROG

(MAGMA) [ n^2 : n in [0..1000]];
(PARI) {a(n) = n^2};
(PARI) b000290(maxn)={for(n=0, maxn, print(n, " ", n^2); )} \\ Anatoly E. Voevudko, Nov 11 2015
(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 */
(Scheme) (define (A000290 n) (* n n)) ;; Antti Karttunen, Oct 06 2017
(Scala) (0 to 59).map(n => n * n) // Alonso del Arte, Oct 07 2019


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), A028338 (2nd diagonal).
A row or column of A132191.
This sequence is related to partitions of 2^n into powers of 2, as it is shown in A002577. So A002577 connects the squares and A000447.  Valentin Bakoev, Mar 03 2009
Boustrophedon transforms: A000697, A000745.
Sequence in context: A331221 A174452 A174902 * A162395 A253909 A305559
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


STATUS

approved



