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%I M4697 N2006 #176 Feb 12 2023 13:30:53
%S 0,1,10,35,84,165,286,455,680,969,1330,1771,2300,2925,3654,4495,5456,
%T 6545,7770,9139,10660,12341,14190,16215,18424,20825,23426,26235,29260,
%U 32509,35990,39711,43680,47905,52394,57155,62196,67525,73150,79079,85320,91881,98770,105995,113564,121485
%N a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.
%C 4 times the variance of the area under an n-step random walk: e.g., with three steps, the 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_, Jul 14 2003
%C Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - _Emeric Deutsch_, May 30 2004
%C 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 07 2004
%C The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. This sequence contains the tetrahedral numbers of A000292 obtained for n= 1,3,5,7,... (see A015219). - _Valentin Bakoev_, Mar 03 2009
%C Using three consecutive numbers u, v, w, (u+v+w)^3-(u^3+v^3+w^3) equals 18 times the numbers in this sequence. - _J. M. Bergot_, Aug 24 2011
%C This sequence is related to A070893 by A070893(2*n-1) = n*a(n)-sum(i=0..n-1, a(i)). - _Bruno Berselli_, Aug 26 2011
%C Number of integer solutions to 1-n <= x <= y <= z <= n-1. - _Michael Somos_, Dec 27 2011
%C Partial sums of A016754. - _Reinhard Zumkeller_, Apr 02 2012
%C Also the number of cubes in the n-th Haüy square pyramid. - _Eric W. Weisstein_, Sep 27 2017
%D G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
%D F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
%D C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
%D L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000447/b000447.txt">Table of n, a(n) for n = 0..1000</a>
%H J. L. Bailey, <a href="/A002309/a002309.pdf">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
%H Valentin Bakoev, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00096-7">Algorithmic approach to counting of certain types m-ary partitions</a>, Discrete Mathematics, Vol. 275 (2004), pp. 17-41. - _Valentin Bakoev_, Mar 03 2009
%H F. E. Croxton and D. J. Cowden, <a href="/A000447/a000447.pdf">Applied General Statistics</a>, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HauyConstruction.html">Haüy Construction</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramid.html">Square Pyramid</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = binomial(2*n+1, 3) = A000292(2*n-1).
%F G.f.: x*(1+6*x+x^2)/(1-x)^4.
%F a(n) = -a(-n) for all n in Z.
%F a(n) = A000330(2*n)-4*A000330(n) = A000466(n)*n/3 = A000578(n)+A007290(n-2) = A000583(n)-2*A024196(n-1) = A035328(n)/3. - _Henry Bottomley_, Jul 14 2003
%F a(n+1) = (2*n+1)*(2*n+2)(2*n+3)/6. - _Valentin Bakoev_, Mar 03 2009
%F a(0)=0, a(1)=1, a(2)=10, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, May 25 2012
%F a(n) = v(n,n-1), where v(n,k) is the central factorial numbers of the first kind with odd indices. - _Mircea Merca_, Jan 25 2014
%F a(n) = A005917(n+1) - A100157(n+1), where A005917 are the rhombic dodecahedral numbers and A100157 are the structured rhombic dodecahedral numbers (vertex structure 9). - _Peter M. Chema_, Jan 09 2016
%F For any nonnegative integers m and n, 8*(n^3)*a(m) + 2*m*a(n) = a(2*m*n). - _Ivan N. Ianakiev_, Mar 04 2017
%F E.g.f.: exp(x)*x*(1 + 4*x + (4/3)*x^2). - _Wolfdieter Lang_, Mar 11 2017
%F a(n) = A002412(n) + A016061(n-1), for n>0. - _Bruce J. Nicholson_, Nov 12 2017
%F From _Amiram Eldar_, Jan 04 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 6*log(2) - 3.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)
%e G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
%e a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
%e a(0) = 0, which corresponds to the empty sum.
%p A000447:=z*(1+6*z+z**2)/(z-1)**4; # _Simon Plouffe_, 1992 dissertation.
%p A000447:=n->n*(4*n^2 - 1)/3; seq(A000447(n), n=0..50); # _Wesley Ivan Hurt_, Mar 30 2014
%t Table[n (4 n^2 - 1)/3, {n, 0, 80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%t LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* _Harvey P. Dale_, May 25 2012 *)
%t Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* _Harvey P. Dale_, Jul 18 2013 *)
%t CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 27 2017 *)
%o (PARI) {a(n) = n * (4*n^2 - 1) / 3};
%o (Haskell)
%o a000447 n = a000447_list !! n
%o a000447_list = scanl1 (+) a016754_list
%o -- _Reinhard Zumkeller_, Apr 02 2012
%o (Maxima) A000447(n):=n*(4*n^2 - 1)/3$ makelist(A000447(n),n,0,20); /* _Martin Ettl_, Jan 07 2013 */
%o (PARI) concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ _Altug Alkan_, Jan 11 2016
%o (Magma) [n*(4*n^2-1)/3: n in [0..50]]; // _Vincenzo Librandi_, Jan 12 2016
%o (Python)
%o def A000447(n): return n*((n**2<<2)-1)//3 # _Chai Wah Wu_, Feb 12 2023
%Y (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 Column 1 in triangles A008956 and A008958.
%Y Cf. A035328, A069072, A190152.
%Y 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. - _Valentin Bakoev_, Mar 03 2009
%Y Cf. A002412, A016061.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
%E Chrystal and Durell references from _R. K. Guy_, Apr 02 2004
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