Search: id:A002412
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%I A002412 M4374 N1839
%S A002412 0,1,7,22,50,95,161,252,372,525,715,946,1222,1547,1925,2360,2856,
%T A002412 3417,4047,4750,5530,6391,7337,8372,9500,10725,12051,13482,15022,
%U A002412 16675,18445,20336,22352,24497,26775,29190,31746,34447,37297,40300
%N A002412 Hexagonal pyramidal numbers, or greengrocer's numbers.
%C A002412 a(n) is the sum of the maximum(m,n) over {(m,n):m,n in positive integers,
               m<=n} [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Oct 
               11 2009]
%D A002412 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002412 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002412 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 194.
%D A002412 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 A002412 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, 
               Vol. 1, No. 4, pp. 323-332, 2002.
%D A002412 I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, 
               in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De 
               Gruyter, 2002.
%H A002412 T. D. Noe, <a href="b002412.txt">Table of n, a(n) for n=0..1000</a>
%H A002412 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002412 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 A002412 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 A002412 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexagonalPyramidalNumber.html">Link to a section of The World of 
               Mathematics.</a>
%F A002412 a(n) = n*(n+1)*(4*n-1)/6. G.f.: x*(1+3*x)/(1-x)^4.
%F A002412 n^3-sum(i^2, i=1..(n-1)) - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de).
%F A002412 Partial sums of n odd triangular numbers, e.g. a(3)=t(1)+t(3)+t(5)=1+6+15=22 
               - Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003
%F A002412 a(n)=sum(i=0, n-1, (n-i)(n+i)) - Jon Perry (perry(AT)globalnet.co.uk), 
               Sep 26 2004
%F A002412 Binomial transform of (1, 6, 9, 4, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 16 2007
%F A002412 a(n) = n*A000292(n) - (n-1)*A000292(n-1) = n*C((n+2),3) - (n-1)*C((n+1),
               3); e.g. a(5) = 95 = 5*35 - 4*20. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 28 2007
%p A002412 seq(add((n^2-k^2),k=0..n),n=0..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 02 2006
%p A002412 a:=n->sum(binomial(n+j,1)*binomial(n-j,1),j=0..n): seq(a(n),n=0..39); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2006
%p A002412 A002412:=z*(1+3*z)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%t A002412 f[n_]:=4*n+1; s1=s2=0;lst={};Do[a=f[n];s1+=a;s2+=s1;AppendTo[lst,s2],
               {n,0,6!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 
               25 2009]
%o A002412 (PARI) v=vector(40,i,t(i)); s=0; forstep(i=1,40,2,s+=v[i]; print1(s",
               "))
%Y A002412 Bisection of A002623. Equals A000578(n)-A000330(n-1).
%Y A002412 Cf. A016061.
%Y A002412 a(n)= A093561(n+2, 3), (4, 1)-Pascal column.
%Y A002412 Cf. A000292.
%Y A002412 Sequence in context: A010001 A014073 A129109 this_sequence A041215 A060822 
               A011926
%Y A002412 Adjacent sequences: A002409 A002410 A002411 this_sequence A002413 A002414 
               A002415
%K A002412 nonn,easy,nice
%O A002412 0,3
%A A002412 N. J. A. Sloane (njas(AT)research.att.com).
%E A002412 Plouffe Maple line edited by R. J. Mathar, May 16 2008

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