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A000447
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a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.
(Formerly M4697 N2006)
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42
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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, 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. 7, 2004.
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]
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
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REFERENCES
| 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]
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.
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.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.
Milan Janjic, Two Enumerative Functions
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| a(n) = (2*n+1)*(2*n+2)(2*n+3)/6. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Mar 03 2009]
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).
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
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EXAMPLE
| 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.
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MAPLE
| A000447:=z*(1+6*z+z**2)/(z-1)**4; [S. Plouffe, 1992 dissertation.]
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MATHEMATICA
| Table[n*(4*n^2-1)/3, {n, 0, 80}] (* From Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
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PROG
| (PARI) {a(n) = n * (4*n^2 - 1) / 3}
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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.
a(n)=A000292(2n-2). A002492(n)=A000292(2n+1).
Column 1 in triangles A008956 and A008958.
Cf. A035328, A069072, A190152.
1) A000447 is a bisection 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]
Sequence in context: A022702 A044468 A109710 * A052472 A049736 A048507
Adjacent sequences: A000444 A000445 A000446 * A000448 A000449 A000450
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999
Chrystal and Durell references from R. K. Guy, Apr 02 2004.
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