Search: id:A002411
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%I A002411 M4116 N1709
%S A002411 0,1,6,18,40,75,126,196,288,405,550,726,936,1183,1470,1800,2176,2601,
%T A002411 3078,3610,4200,4851,5566,6348,7200,8125,9126,10206,11368,12615,13950,
%U A002411 15376,16896,18513,20230,22050,23976,26011,28158,30420,32800,35301
%N A002411 Pentagonal pyramidal numbers: n^2*(n+1)/2.
%C A002411 a(n)=n^2(n+1)/2 is half the number of colorings of three points on a 
               line with n+1 colors. - Ron Hardin (rhhardin(AT)att.net), Feb 23 
               2002
%C A002411 a(n) = number of (n+6)-bit binary sequences with exactly 7 1's none of 
               which is isolated. A 1 is isolated if its immediate neighbor(s) are 
               0. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
%C A002411 Also as a(n)=(1/6)*(3*n^3+3*n^2), n>0: structured trigonal prism numbers 
               (Cf. A100177 - structured prisms; A100145 for more on structured 
               numbers). - James A. Record (james.record(AT)gmail.com), Nov. 7, 
               2004.
%C A002411 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Nov 18 2005
%C A002411 If Y is a 3-subset of an n-set X then, for n>=5, a(n-4) is the number 
               of 5-subsets of X having at least two elements in common with Y. 
               - Milan R. Janjic (agnus(AT)blic.net), Nov 23 2007
%C A002411 a(n-1), n>=2, is the number of ways to have n identical objects in m=2 
               of alltogether n distinguishable boxes (n-2 boxes stay empty). W. 
               Lang, Nov 13 2007.
%C A002411 a(n+1) is the convolution of (n+1) and (3n+1). [From Paul Barry (pbarry(AT)wit.ie), 
               Sep 18 2008]
%C A002411 A002411[n+1]=denom(2/((n+2)*(n+1)^2)) [From Stephen Crowley (crow(AT)crowlogic.net), 
               Jun 28 2009]
%D A002411 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002411 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002411 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 194.
%D A002411 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 
               256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see 
               vol. 2, p. 2.
%D A002411 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer 
               Press, NY, 1950, p. 36.
%D A002411 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, 
               Chem. Ber. 30 (1897), 1917-1926.
%D A002411 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see 
               p. 166, Table 10.4/I/5).
%H A002411 T. D. Noe, <a href="b002411.txt">Table of n, a(n) for n=0..1000</a>
%H A002411 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A002411 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002411 P. Chinn and S. Heubach, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Integer Sequences Related to Compositions without 
               2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A002411 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 A002411 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 A002411 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PentagonalPyramidalNumber.html">Link to a section of The World of 
               Mathematics.</a>
%H A002411 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WienerIndex.html">Wiener Index</a>
%F A002411 Average of n^2 and n^3.
%F A002411 a(n) = sum of n smallest multiples of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Sep 20 2002
%F A002411 G.f.: x(1+2x)/(1-x)^4 - Paul Barry (pbarry(AT)wit.ie), Mar 21 2003
%F A002411 a(n)=sum{k=0..n, n(n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
%F A002411 a(n) = whole numbers * triangular numbers - Xavier Acloque Oct 27 2003
%F A002411 a(n) = (1/2)*Sum[Sum[(i+j),{i, 1, n}],{j, 1, n}] = (1/2)*(n^2+n^3) = 
               (1/2)*A011379(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Apr 13 2006
%F A002411 sum (n*j,j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 
               12 2006
%F A002411 a(n)=sum(sum(k, j=1..n),k=1..n), n>=0 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 11 2007
%F A002411 Row sums of triangle A127739, triangle A132118; and binomial transform 
               of [1, 5, 7, 3, 0, 0, 0,...] = (1, 6, 18, 40, 75,...). - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Aug 10 2007
%F A002411 G.f.: x*F(2,3;1;x); [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2008]
%F A002411 sum(1/A002411[j],j=1..infinity)=hypergeom([1, 1, 1], [ 2, 3], 1)=-2+2*Zeta(2) 
               [From Stephen Crowley (crow(AT)crowlogic.net), Jun 28 2009]
%e A002411 a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable 
               boxes in binomial(4,2)*(2!/(1!*1!)+ 2!/(2!) ) = 6*(2+1) =18 ways. 
               The m=2 part partitions of 4, namely (1,3) and (2^2) specify the 
               filling of each of the 6 possible two box choices. W. Lang, Nov 13 
               2007.
%p A002411 Epi:=(r,n)->stirling2(r,n): [seq (Epi(n+1,n),n=0..41)]; - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Dec 05 2006
%p A002411 seq((n-1)*binomial(n,2), n=1..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 28 2007
%p A002411 a:=n->sum(sum(k, j=1..n),k=1..n): seq(a(n), n=0..41); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 11 2007
%p A002411 A002411:=(1+2*z)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%p A002411 a:=n->(sum((numbcomp(n,3)), j=3..n)):seq(a(n), n=2..37); [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
%t A002411 lst={};Do[s0=n^2;s1=n^3;AppendTo[lst,(s1+s0)/2],{n,0,5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Feb 19 2009]
%t A002411 Table[Sum[Binomial[n, 2], {i, 2, n}], {n, 1, 41}] [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
%o A002411 (PARI) a(n)=n^2*(n+1)/2
%Y A002411 Cf. A015223, A015224, A014799, A014800.
%Y A002411 Cf. A011379, A127739, A132118.
%Y A002411 A006002(n)=-a(-1-n).
%Y A002411 a(n)= A093560(n+2, 3), (3, 1)-Pascal column.
%Y A002411 A row or column of A132191.
%Y A002411 Second column of triangle A103371.
%Y A002411 Sequence in context: A129863 A035489 A122061 this_sequence A023658 A059834 
               A015224
%Y A002411 Adjacent sequences: A002408 A002409 A002410 this_sequence A002412 A002413 
               A002414
%K A002411 nonn,easy,nice
%O A002411 0,3
%A A002411 N. J. A. Sloane (njas(AT)research.att.com).
%E A002411 zeta hypergeometric formula [From Stephen Crowley (crow(AT)crowlogic.net), 
               Jun 28 2009]

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