A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.

0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• changed

Comments

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

Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch, May 30 2004

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

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

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

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

Number of integer solutions to 1-n <= x <= y <= z <= n-1. - Michael Somos, Dec 27 2011

Partial sums of A016754. - Reinhard Zumkeller, Apr 02 2012

Also the number of cubes in the n-th Haüy square pyramid. - Eric W. Weisstein, Sep 27 2017

Examples

			G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
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.
a(0) = 0, which corresponds to the empty sum.
		

References

• G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
• F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
• C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
• L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
• Valentin Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, Vol. 275 (2004), pp. 17-41. - _Valentin Bakoev_, Mar 03 2009
• F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
• Milan Janjic, Two Enumerative Functions.
• T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Eric Weisstein's World of Mathematics, Haüy Construction.
• Eric Weisstein's World of Mathematics, Square Pyramid.
• Index entries for two-way infinite sequences.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

(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.

Column 1 in triangles A008956 and A008958.

Cf. A035328, A069072, A190152.

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

Cf. A002412, A016061.

Programs

Haskell

a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
-- Reinhard Zumkeller, Apr 02 2012

Magma

[n*(4*n^2-1)/3: n in [0..50]]; // Vincenzo Librandi, Jan 12 2016

Maple

A000447:=z*(1+6*z+z**2)/(z-1)**4; # Simon Plouffe, 1992 dissertation.
A000447:=n->n*(4*n^2 - 1)/3; seq(A000447(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Mathematica

Table[n (4 n^2 - 1)/3, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

Maxima

A000447(n):=n*(4*n^2 - 1)/3$ makelist(A000447(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

PARI

{a(n) = n * (4*n^2 - 1) / 3};

PARI

concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016

Python

def A000447(n): return n*((n**2<<2)-1)//3 # Chai Wah Wu, Feb 12 2023

Formula

a(n) = binomial(2*n+1, 3) = A000292(2*n-1).

G.f.: x*(1+6*x+x^2)/(1-x)^4.

a(n) = -a(-n) for all n in Z.

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

a(n+1) = (2*n+1)*(2*n+2)(2*n+3)/6. - Valentin Bakoev, Mar 03 2009

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

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

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

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

E.g.f.: exp(x)*x*(1 + 4*x + (4/3)*x^2). - Wolfdieter Lang, Mar 11 2017

a(n) = A002412(n) + A016061(n-1), for n>0. - Bruce J. Nicholson, Nov 12 2017

From Amiram Eldar, Jan 04 2022: (Start)

Sum_{n>=1} 1/a(n) = 6*log(2) - 3.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)

Extensions

Chrystal and Durell references from R. K. Guy, Apr 02 2004