0,3

Figurate numbers based on 4-dimensional regular convex polytope called the 4-measure polytope, 4-hypercube or tessaract with Schlaefli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004

Sum(k>0,1/a(k))=Pi^4/90 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 20 2009]

Totally multiplicative sequence with a(p) = p^4 for prime p. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 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).

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.

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

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

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.

H. Bottomley, Illustration of initial terms

H. Bottomley, Some Smarandache-type multiplicative sequences

Hyun Kwang Kim, On Regular Polytope Numbers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for "core" sequences

Multiplicative with a(p^e) = p^(4e). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f.: x*(1+11*x+11*x^2+x^3)/(1-x)^5. More generally, g.f. for n^m is Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292).

Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, Sep 11 2005.

E.g.f.: (x+7x^2+6x^3+x^4)*e^x. More generally, the general form for the e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial of order n. - Franklin T. Adams-Watters, Sep 11 2005.

a(n)=sum(sum(sum(n, j=1..n),k=1..n),m=1..n), n>=0 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007

a(n) = {least common multiple of n and (n-1)^3}-(n-1)^3. E.g.: {least common multiple of 1 and (1-1)^3}-(1-1)^3 = 0, {least common multiple of 2 and (2-1)^3}-(2-1)^3 = 1, {least common multiple of 3 and (3-1)^3}-(3-1)^3 = 16, {least common multiple of 4 and (4-1)^3}-(4-1)^3 = 81, ... - Mats Granvik (mgranvik(AT)abo.fi), Sep 24 2007

a(n) = C(n+3,4) + 11 C(n+2,4) + 11 C(n+1,4) + C(n,4)

A000583 := n->n^4;

a:=n->sum(sum(n^2, j=1..n), k=1..n): seq(a(n), n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007

a:=n->sum(sum(sum(n, j=1..n), k=1..n), m=1..n): seq(a(n), n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007

A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; [S. Plouffe in his 1992 dissertation. Gives sequence without initial zero.]

with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 25 2008

Cf. A000538, A005917.

Cf. A000332, A014820, A092181, A092182, A092183.

a(n) = A123865(n) + 1.

Adjacent sequences: A000580 A000581 A000582 this_sequence A000584 A000585 A000586

Sequence in context: A017672 A055013 A080150 this_sequence A050751 A014188 A050463

nonn,core,easy,nice,mult,new

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

More terms from Mats Granvik (mgranvik(AT)abo.fi), Sep 24 2007

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009