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A000583
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Fourth powers: a(n) = n^4.
(Formerly M5004 N2154)
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93
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0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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]
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REFERENCES
| 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
H. Bottomley, Illustration of initial terms
H. Bottomley, Some Smarandache-type multiplicative sequences
Ralph Greenberg, Math for Poets
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Hyun Kwang Kim, On Regular Polytope Numbers
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.
Eric Weisstein's World of Mathematics, Biquadratic Number
Index entries for "core" sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| 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)
a(n) = n*A177342(n) - sum [i = 1..n-1] A177342(i) - (n-1), with n>0 . For n=5, a(5) = 5*A177342(5) - sum [i=1..4] A177342(i) - 4 = 5*149-(75+31+9+1)-4 = 625 [From Bruno Berselli (berselli.bruno(AT)yahoo.it), May 07 2010]
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MAPLE
| 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
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MATHEMATICA
| Range[0, 100]^4 (*From Vladimir Joseph Stephan Orlovsky, Mar 14 2011*)
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PROG
| (PARI) A000583(n) = n^4 [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 09 2009]
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CROSSREFS
| Cf. A000538, A005917, A000332, A014820, A092181, A092182, A092183.
a(n) = A123865(n) + 1.
Sequence in context: A017672 A055013 A080150 * A050751 A014188 A050463
Adjacent sequences: A000580 A000581 A000582 * A000584 A000585 A000586
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KEYWORD
| nonn,core,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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
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