Search: id:A000447
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%I A000447 M4697 N2006
%S A000447 0,1,10,35,84,165,286,455,680,969,1330,1771,2300,2925,3654,4495,5456,
%T A000447 6545,7770,9139,10660,12341,14190,16215,18424,20825,23426,26235,29260,
%U A000447 32509,35990,39711,43680,47905,52394,57155,62196,67525,73150,79079
%N A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4n^2 - 1)/3.
%C A000447 4 times variance of the area under an n step random walk: e.g. with three 
               steps, area can be 9/2, 7/2, 3/2, 1/2, -1/2, -3/2, -7/2, or -9/2 
               each with probability 1/8, giving a variance of 35/4 or a(3)/4. - 
               Henry Bottomley (se16(AT)btinternet.com), Jul 14 2003
%C A000447 Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), May 30 2004
%C A000447 Also a(n)=(1/6)*(8*n^3-2*n), n>0: structured octagonal diamond numbers 
               (vertex structure 9) (Cf. A059722 = alternate vertex; A000447 = structured 
               diamonds); and structured tetragonal anti-diamond numbers (vertex 
               structure 9) (Cf. A096000 = alternate vertex; A100188 = structured 
               anti-diamonds). Cf. A100145 for more on structured numbers. - James 
               A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A000447 The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. A000447 
               contains the tetrahedral numbers obtained for n= 1,3,5,7,... [From 
               Valentin Bakoev (v_bakoev(AT)yahoo.com), Mar 03 2009]
%D A000447 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000447 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000447 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]
%D A000447 G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. 
               XX, Sect. 10, Example 2.
%D A000447 F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., 
               Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
%D A000447 C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise 
               IIIe, No. 4.
%D A000447 L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
%D A000447 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (11).
%D A000447 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%H A000447 T. D. Noe, <a href="b000447.txt">Table of n, a(n) for n=0..1000</a>
%H A000447 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A000447 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 A000447 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 A000447 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A000447 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000447 V. Bakoev, <a href="http://www.uni-vt.bg/showfullpub.asp?u=32&fn=AACCTMP.pdf">
               Algorithmic approach to counting of certain types m-ary partitions</
               a>, Discrete Mathematics, 275 (2004) pp. 17-41.
%F A000447 a(n)=binomial(2*n+1, 3)=A000292(2*(n-1))
%F A000447 G.f.: x(1+6x+x^2)/(1-x)^4. a(-n)=-a(n).
%F A000447 a(n) = A000330(2n)-4*A000330(n) = A000466(n)*n/3 = A000578(n)+A007290(n-2) 
               = A000583(n)-2*A024196(n-1) = A035328(n)/3. - Henry Bottomley (se16(AT)btinternet.com), 
               Jul 14 2003
%F A000447 a(n)= (2n+1)(2n+2)(2n+3)/6, for n= 0,1,2,3,... [From Valentin Bakoev 
               (v_bakoev(AT)yahoo.com), Mar 03 2009]
%p A000447 A000447:=z*(1+6*z+z**2)/(z-1)**4; [S. Plouffe, 1992 dissertation.]
%t A000447 s = 0; lst = {s}; Do[s += n^2; AppendTo[lst, s], {n, 1, 80, 2}]; lst 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
%o A000447 (PARI) a(n)=n*(4*n^2-1)/3
%Y A000447 (1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, 
               A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, 
               A007588, A062025, A063521, A063522, A063523.
%Y A000447 a(n)=A000292(2n-2). A002492(n)=A000292(2n+1).
%Y A000447 Column 1 in triangles A008956 and A008958.
%Y A000447 Cf. A035328, A069072.
%Y A000447 1) A000447 is a subsequence of A000292 (the tetrahedral numbers). The 
               members of A000447 take the odd places in A000292; 2) A000447 is 
               related to partitions of 2^n into powers of 2, as it is shown in 
               the formula, example and cross-references of A002577. So A002577 
               relates A000447 and A000290. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), 
               Mar 03 2009]
%Y A000447 Sequence in context: A022702 A044468 A109710 this_sequence A052472 A049736 
               A048507
%Y A000447 Adjacent sequences: A000444 A000445 A000446 this_sequence A000448 A000449 
               A000450
%K A000447 easy,nonn,nice
%O A000447 0,3
%A A000447 N. J. A. Sloane (njas(AT)research.att.com).
%E A000447 More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), 
               Dec 29 1999
%E A000447 Chrystal and Durell references from R. K. Guy, Apr 02 2004.

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