0,3

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

Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 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]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

T. D. Noe, Table of n, a(n) for n=0..1000

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

V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

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

G.f.: x(1+6x+x^2)/(1-x)^4. a(-n)=-a(n).

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

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]

A000447:=z*(1+6*z+z**2)/(z-1)**4; [S. Plouffe, 1992 dissertation.]

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]

(PARI) a(n)=n*(4*n^2-1)/3

(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.

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]

Sequence in context: A022702 A044468 A109710 this_sequence A052472 A049736 A048507

Adjacent sequences: A000444 A000445 A000446 this_sequence A000448 A000449 A000450

easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

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.