Search: id:A002378
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%I A002378 M1581 N0616
%S A002378 0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,
%T A002378 462,506,552,600,650,702,756,812,870,930,992,1056,1122,1190,1260,1332,
%U A002378 1406,1482,1560,1640,1722,1806,1892,1980,2070,2162,2256,2352,2450,2550
%N A002378 Oblong (or promic, pronic, or heteromecic) numbers: n(n+1).
%C A002378 4*a(n)+1 are the odd squares A016754(n).
%C A002378 The word "pronic" (used by Dickson) is incorrect. - Michael Somos. According 
               to the 2nd edition of Webster, the correct word is "promic" - R. 
               K. Guy (rkg(AT)cpsc.ucalgary.ca)
%C A002378 a(n) is the number of minimal vectors in the root lattice A_n (see Conway 
               and Sloane, p. 109).
%C A002378 Let M_n denotes the n X n matrix M_n(i,j)=(i+j); then the characteristic 
               polynomial of M_n is x^(n-2) * (x^2-a(n)*x - A002415(n)). - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Nov 09 2002
%C A002378 The greatest LCM of all pairs (j,k) for j<k=<n for n>1. - Robert G. Wilson 
               v Jun 19 2004.
%C A002378 First differences are 2 4 6 8 10 12 14... (whilst first differences of 
               the squares are 1 3 5 7 9 11 13...) - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), 
               Dec 29 2005
%C A002378 25 appended to these numbers corresponds to squares of numbers ending 
               in 5 (i.e. to squares of A017329). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Mar 24 2006
%C A002378 Number of circular binary words of length n+1 having exactly one occurrence 
               of 01. Example: a(2)=6 because we have 001, 010, 011, 100, 101 and 
               110. Column 1 of A119462. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 21 2006
%C A002378 The sequence of iterated square roots sqrt(N+sqrt(N+...)) has for N=1,
               2,... the limit (1+sqrt(1+4*N))/2. For N=a(n) this limit is n+1, 
               n=1,2,.... For all other numbers N, N>=1, this limit is not a natural 
               number. Examples: n=1, a(1)=2: sqrt(2+sqrt(2+ ...)) = 1+1 =2; n=2, 
               a(2)=6: sqrt(6+sqrt(6+ ...)) = 1+2 =3. W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               May 05 2006.
%C A002378 Nonsquare integers m divisible by ceil(sqrt(m)), except m=0. - Max Alekseyev 
               (maxale(AT)gmail.com), Nov 27 2006
%C A002378 a(n) = 2*sum(1..n-1). - Artur Jasinski (grafix(AT)csl.pl), Jan 09 2007
%C A002378 The number of off-diagonal elements of an n+1 X n+1 matrix. - Artur Jasinski 
               (grafix(AT)csl.pl), Jan 11 2007
%C A002378 a(n) is equal to the number of functions f:{1,2}->{1,2,...,n+1} such 
               that for a fixed x in {1,2} and a fixed y in {1,2,...,n+1} we have 
               f(x)<>y. - Aleksandar M. Janjic and Milan R. Janjic (agnus(AT)blic.net), 
               Mar 13 2007
%C A002378 Numbers m>=0 such that round(sqrt(m+1))-round(sqrt(m))=1. - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
%C A002378 Numbers m>=0 such that ceiling(2*sqrt(m+1))-1=1+floor(2*sqrt(m)). - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
%C A002378 Numbers m>=0 such that fract(sqrt(m+1))>1/2 and fract(sqrt(m))<1/2 where 
               fract(x) is the fractional part (i.e. fract(x)=x-floor(x), x>=0). 
               - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007
%C A002378 Sequence allows us to find X values of the equation: 4*X^3 + X^2 = Y^2. 
               To find Y values: b(n)=n(n+1)(2n+1). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               Nov 06 2007
%C A002378 Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized 
               factorials" comprised of a(n) are given by the elements of the Lah 
               matrix, unsigned A111596, e.g., a(1)*a(2)*a(3)/ 3! = -A111596(4,1) 
               = 24 . - Tom Copeland (tcjpn(AT)msn.com), Nov 20 2007
%C A002378 If Y is a 2-subset of an n-set X then, for n>=2, a(n-2) is the number 
               of 2-subsets and 3-subsets of X having exactly one element in common 
               with Y. - Milan R. Janjic (agnus(AT)blic.net), Dec 28 2007
%C A002378 Infinite Sum[1/a[n+1],{n,1,Infinity}] = 1 but series is very slowly convergent. 
               [From Artur Jasinski (grafix(AT)csl.pl), Sep 28 2008]
%C A002378 a(n)=A061037(4n)=(n+1/2)^2-1/4=((2n+1)^2-1)/4.From Balmer spectrum of 
               hydrogen. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 03 2008]
%C A002378 a(n) coincides with the vertex of a parabola of even width in the Redheffer 
               matrix, directed toward zero. An integer p is prime iff for all integer 
               k, the parabola y = kx - x^2 has no integer solution with 1 < x < 
               k when y = p; a(n) corresponds to odd k. [From Reikku Kulon (reikku(AT)gmail.com), 
               Nov 30 2008]
%C A002378 Except for the first term of [A002378], if X=[A144396], Y=[A007395], 
               A= [A002378], we have, for all other terms, Pell's equation: [A144396]^2 
               - [A002378]*[A007395]^2=1; (X^2-A*Y^2=1); example: 3^2-2*2^2=1; 5^2-6*2^2=1; 
               19^2-90*2^2=1, and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%C A002378 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 21 
               2009: (Start)
%C A002378 The third differences of certain values of the hypergeometric function 
               3F2 lead to the squares of the oblong numbers i.e. 3F2([1,n+1,n+1], 
               [n+2,n+2], z=1) - 3*3F2([1,n+2,n+2], [n+3,n+3], z=1) + 3*3F2([1,n+3,
               n+3], [n+4,n+4], z=1) - 3F2([1,n+4,n+4], [n+5,n+5], z=1) = (1/((n+2)*(n+3)))^2 
               for n = -1, 0, 1,2, .. . See also A162990.
%C A002378 (End)
%C A002378 Number of units of a(n) belongs to a periodic sequence: 0, 2, 6, 2, 0. 
               [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
%C A002378 a(A007018(n)) = A007018(n+1), see sequence A007018 (1,2,6,42,1806,...), 
               i.e. A007018(n+1) = A007018(n) th oblong numbers. [From Jaroslav 
               Krizek (jaroslav.krizek(AT)atlas.cz), Sep 13 2009]
%C A002378 a(j) = number of non-zero values of floor (j^2/n) taken over all n >= 
               1 for each j, with 1 <= j <= n-1.
%C A002378 Except for the first term, a(n)=2*n+a(n-1), (with a(1)=2) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]
%C A002378 Number of permutations of three distinct letters (ABC) 0 to n times ("-", 
               ABC, AABBCC, AAABBBCCC, etc...)and one after the other to resemble 
               motif:(n+1)* A, n*B, and (n-1)*C, example: ("-"),AAB (2-1-0), AAABBC 
               (3-2-1), AAAABBBCC (4-3-2), AAAAABBBBCCC (5-4-3), etc... (m-2) fixed 
               point. Example:if "-" and motif "-" then 0*(m-2) fixed point, if 
               ABC (1-1-1)and motif AAB (2-1-0) then 2*(m-2) fixed point, if AABBCC 
               and motif AAABBC then 6*(m-2) fixed point, ..., if AAAAABBBBBCCCCC 
               (5-5-5) and motif AAAAAABBBBBCCCC (6-5-4) then 30*(m-2) fixed point, 
               etc...(m=three n). [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 07 2009]
%D A002378 W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open 
               Court, page 67.
%D A002378 J. H. Conway and R. K. Guy, The Book of Numbers, 1996, p. 34.
%D A002378 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag.
%D A002378 L. E. Dickson, History of the Theory of Numbers, Vol. 1: Divisibility 
               and Primality. New York: Chelsea, p. 357, 1952.
%D A002378 L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine 
               Analysis. New York: Chelsea, pp. 6, 232-233, 350 and 407, 1952.
%D A002378 H. Eves, An Introduction to the History of Mathematics, revised, Holt, 
               Rinehart and Winston, 1964, page 72.
%D A002378 Nicomachus of Gerasa, Introduction to Arithmetic, translation by Martin 
               Luther D'Ooge, Ann Arbor, University of Michigan Press, 1938, p. 
               254.
%D A002378 C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford 
               University Press, 1966, pp. 61-62.
%D A002378 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002378 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002378 F. J. Swetz, From Five Fingers to Infinity, Open Court, 1994, p. 219.
%D A002378 R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 
               of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett 
               et al., Peters, 2003.
%H A002378 T. D. Noe, <a href="b002378.txt">Table of n, a(n) for n = 0..1000</a>
%H A002378 H. Bottomley, <a href="a2378.gif">Illustration of initial terms of A000217, 
               A002378</a>
%H A002378 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=410">
               Encyclopedia of Combinatorial Structures 410</a>
%H A002378 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A002378 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 A002378 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 A002378 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
               Geometri/Pronique.htm">Nombres Proniques</a>
%H A002378 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PronicNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A002378 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LeibnizHarmonicTriangle.html">Leibniz Harmonic Triangle</a>
%H A002378 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CrownGraph.html">Crown Graph</a>
%H A002378 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WienerIndex.html">Wiener Index</a>
%H A002378 Wolfram Research, <a href="http://functions.wolfram.com/webMathematica/
               FunctionEvaluation.jsp?name=Hypergeometric3F2">Hypergeometric Function 
               3F2</a>, The Wolfram Functions site. [From Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Jul 21 2009]
%H A002378 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A002378 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A002378 G.f.: (2*x)/(1-x)^3; a(n)=a(n-1)+2*n, a(0)=0.
%F A002378 Sum_{n >= 1} n*(n+1) = n(n+1)(n+2)/3 (cf. A007290).
%F A002378 Sum_{n >= 1} 1/(n*(n+1)) = 1. (Cf. Tijdeman)
%F A002378 1 = 1/2 + Sum(n = 1 through infinity) 1/[2*a(n)] = 1/2 + 1/4 + 1/12 + 
               1/24 + 1/40 + 1/60...with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 
               11/12, 13/14... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 16 
               2003
%F A002378 a(n)*a(n+1)=a(n*(n+2)); e.g. a(3)*a(4)=12*20=240=a(3*5) - Charlie Marion 
               (charliemath(AT)verizon.net), Dec 29 2003
%F A002378 Sum_{k=1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v Feb 04 2005.
%F A002378 a(n)=A046092/2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 
               08 2006
%F A002378 Log 2 = Sum(n=0, inf.) 1/a(2n+1)= 1/2 + 1/12 + 1/30 + 1/56 + 1/90...; 
               = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) ...= Sum(n=0, 
               inf.): (-1)^n/(Nn+1), with N=1. Log 2 = Integral_{0..1} 1/(1+x) dx 
               = .69314718...; sum: 1/2 + 1/12 + 1/30 + 1/56 + 1/90 = 1627/2520 
               = .64563... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 22 2003
%F A002378 a(n)=A049598-A124080; a(n)=A124080-A033996: a(n)=A033996-A028896: a(n)=A028896-A046092. 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 06 2007
%F A002378 a(n-1)=n^2-n = A000290(n)-A000027(n) for n>=1. a(n) = inverse (frequency 
               distribution) sequence of A000194(n).- Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Jul 26 2007
%F A002378 (2, 6, 12, 20, 30,...) = binomial transform of (2, 4, 2). - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Nov 28 2007
%F A002378 a(n)=A000217(n)*2. - Omar E. Pol (info(AT)polprimos.com), May 14 2008
%F A002378 a(n) = A006503(n) - A000292(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 24 2008]
%F A002378 a(0):=0, a(n)=a(n-1)+1+floor(x), where x is the minimal positive solution 
               to fract(sqrt(a(n-1)+1+x))=1/2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Dec 31 2008
%F A002378 E.g.f.:(x+2)*x*exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443), 
               Feb 06 2009]
%F A002378 Product_{i=2..infinity} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/
               A000796. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 
               2009]
%F A002378 egf: ((-x+1)*ln(-x+1)+x)/x^2 also int(((-x+1)*ln(-x+1)+x)/x^2,x=0..1)=Zeta(2)-1 
               [From Stephen Crowley (crow(AT)crowlogic.net), Jul 11 2009]
%F A002378 a(n)=2*n+a(n-1)-2 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 29 2009]
%e A002378 For n=2, a(2)=2*2+0-2=2; n=3, a(3)=2*3+2-2=6; n=4, a(4)=2*4+6-2=12 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]
%p A002378 [ seq(n*(n+1), n=0..100) ];
%p A002378 [seq(2*binomial(n,2),n=1..51)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 24 2006
%p A002378 [seq(numbperm (n, 2), n=1..51)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 25 2007
%p A002378 a:=n->sum(numbcomb (n,1), j=0..n): seq(a(n), n=0..50); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 25 2007
%p A002378 a:=n->sum(n+2*j, j=0..n)/2: seq(a(n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 29 2007
%p A002378 seq(sum(sum(gcd(k,j),j=1..n), k=0..n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 01 2007
%p A002378 A002378:=-2*z/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A002378 with (combinat):seq(fibonacci(3,n)+n-1, n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 07 2008
%p A002378 with(finance):seq(add(futurevalue( k,3,2),k=0..n)/8,n=0..50); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008
%p A002378 with(finance):seq(add(cashflows([2,k,k], 0 ),k=0..n),n=-1..51); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008
%p A002378 restart: G(x):=x^2*exp(x): f[0]:=G(x): for n from 1 to 51 do f[n]:=diff(f[n-1],
               x) od: x:=0: seq(f[n],n=1..51);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 05 2009]
%t A002378 Table[ n(n + 1), {n, 0, 50}] (from Robert G. Wilson v Jun 19 2004)
%t A002378 Table[(n^2 - n), {n, 51}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 21 2007
%o A002378 (PARI) je=[]; for(n=0,5000, if(issquare(4*n+1),je=concat(je,n))); je
%o A002378 sage: [lucas_number1(3,n,n) for n in xrange(1,52)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 16 2008
%o A002378 (PARI) j=[];for (n=0,40,j=concat(j,n^2+floor(((n+1)^2 - n^2)/2)));j [From 
               Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 09 2008]
%Y A002378 Partial sums of A005843 (even numbers). Twice triangular numbers A000217. 
               Partial sums give A007290.
%Y A002378 1/beta(n, 2) in A061928.
%Y A002378 a(n) = A110660(2*n).
%Y A002378 Cf. A035106, A087811, A119462, A127235.
%Y A002378 Cf. A049598, A124080, A033996, A028896, A046092.
%Y A002378 Cf. A000217, A005563, A046092, A001082.
%Y A002378 Cf. A059300, A059297, A059298 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Aug 06 2008]
%Y A002378 Cf. A059300, A059297, A059298.
%Y A002378 Cf. A000040 [From Reikku Kulon (reikku(AT)gmail.com), Nov 30 2008]
%Y A002378 Cf. A000217.
%Y A002378 Cf. A144396, A007395 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%Y A002378 Cf. A166373
%Y A002378 Sequence in context: A160942 A160929 A103505 this_sequence A005991 A003274 
               A121315
%Y A002378 Adjacent sequences: A002375 A002376 A002377 this_sequence A002379 A002380 
               A002381
%K A002378 nonn,easy,core,nice,new
%O A002378 0,2
%A A002378 N. J. A. Sloane (njas(AT)research.att.com).
%E A002378 Additional comments from Michael Somos
%E A002378 Corrected l.h.s. of my formula - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 15 2009
%E A002378 Comment and cross-reference added by Christopher Hunt Gribble (chris.eveswell(AT)virgin.net), 
               Oct 13 2009

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