0,3

Zero followed by partial sums of A005408 (odd numbers). - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), 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 (blekraj(AT)yahoo.com), 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 (blekraj(AT)yahoo.com), Jun 30 2004. See also A000037.

First sequence ever computed by electronic computer, on EDSAC, May 6 1949 (see Renwick link). - Russ Cox (rsc(AT)swtch.com), Apr 20 2006

Numbers n such that the imaginary quadratic field Q[Sqrt[ -n]] has four units. - Marc LeBrun (mlb(AT)well.com), Apr 12 2006

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

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

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

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

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 (ctibor.zizka(AT)seznam.cz), 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 (qntmpkt(AT)yahoo.com), Aug 17 2008]

Equals row sums of triangles A143595 and A056944 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008]

Number of divisors of 6^n - J. Lowell (jhbubby(AT)mindspring.com), Aug 30 2008

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]

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]

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 (sriperso(AT)gmail.com), Jun 25 2009]

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]

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]

Totally multiplicative sequence with a(p) = p^2 for prime p. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 2009]

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. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Mar 03 2009]

R. P. Burn & A. Chetwynd, A Cascade Of Numbers, "The prison door problem" Problem 4 pp. 5-7;79-80 Arnold London 1996.

M. Gardner, Time Travel and Other Mathematical Bewilderments, Chapter 6 pp. 71-2, W.H.Freeman NY 1988.

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

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.

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

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

S. 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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

D. Surendran, Chimbumu and Chickwama get out of jail

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

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

Multiplicative with a(p^e) = p^(2e). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

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).

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

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

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

For n>0: a(n) = A130064(n)*A130065(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 05 2007

a(n) = Sum(A002024(n,k): 1<=k<=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 24 2007

Left edge of the triangle in A132111: a(n)=A132111(n,0). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 10 2007

a(n) = {least common multiple of n and n-1} - (n-1). - Mats Granvik (mgranvik(AT)abo.fi), Sep 16 2007

Binomial transform of [1, 3, 2, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2007

a(n) = C(n+1,2) + C(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 (pevnev(AT)juno.com), May 18 2008

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]

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]

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]

Product_{i=2..infinity} (1-2/a(i)) = -sin(A063448)/A063448. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 2009]

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]

G.f.: x*(1-x)/exp(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]

a(n) = A002378(n-1) + n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 14 2009]

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=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]

A000290 := n->n^2;

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

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

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]

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]

a[n_] := n^2; Table[a[n], {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 30 2006

Array[ #^2 &, 60, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006

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]

s=0; lst={}; Do[s+=n; AppendTo[lst, s], {n, 1, 5!, 2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]

(MAGMA) [ n^2 : n in [0..1000]];

(PARI) a(n)=n^2

(Other) sage: [log(e^(n^2))for n in xrange(0, 34)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]

Cf. A092205, A128200, A005408, A128201, A002522, A005563, A008865, A059100, A143051, A143470, A143595, A056944.

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 [From David Scheers (dscheers(AT)webpoint.nl), Mar 15 2009]

A004159, A159918. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 25 2009]

Sequence in context: A106545 A093837 A069821 this_sequence A162395 A144913 A018885

Adjacent sequences: A000287 A000288 A000289 this_sequence A000291 A000292 A000293

nonn,core,easy,nice,mult,new

N. J. A. Sloane (njas(AT)research.att.com).