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

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

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; [Conjectured by S. Plouffe in his 1992 dissertation.]

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.

Sequence in context: A017672 A055013 A080150 this_sequence A050751 A014188 A050463

Adjacent sequences: A000580 A000581 A000582 this_sequence A000584 A000585 A000586

nonn,core,easy,nice,mult

njas

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