A000583 Fourth powers: a(n) = n^4. (Formerly M5004 N2154)

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

Offset

0,3

Comments

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

Totally multiplicative sequence with a(p) = p^4 for prime p. - Jaroslav Krizek, Nov 01 2009

The binomial transform yields A058649. The inverse binomial transforms yields the (finite) 0, 1, 14, 36, 24, the 4th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013

Generate Pythagorean triangles with parameters a and b to get sides of lengths x = b^2-a^2, y = 2*a*b, and z = a^2 + b^2. In particular use a=n-1 and b=n for a triangle with sides (x1,y1,z1) and a=n and b=n+1 for another triangle with sides (x2,y2,z2). Then x1*x2 + y1*y2 + z1*z2 = 8*a(n). - J. M. Bergot, Jul 22 2013

For n > 0, a(n) is the largest integer k such that k^4 + n is a multiple of k + n. Also, for n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n^2. - Derek Orr, Sep 04 2014

Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017

a(n+2)/2 is the area of a trapezoid with vertices at (T(n), T(n+1)), (T(n+1), T(n)), (T(n+1), T(n+2)), and (T(n+2), T(n+1)) with T(n)=A000292(n) for n >= 0. - J. M. Bergot, Feb 16 2018

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).

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).

Links

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

Henry Bottomley, Illustration of initial terms

Henry Bottomley, Some Smarandache-type multiplicative sequences

Ralph Greenberg, Math for Poets.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.

Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.

Eric Weisstein's World of Mathematics, Biquadratic Number.

Index entries for "core" sequences

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Index entries for sequences related to Benford's law

Formula

a(n) = A123865(n)+1 = A002523(n)-1.

Multiplicative with a(p^e) = p^(4e). - David W. Wilson, 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 + 7*x^2 + 6*x^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

Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - Jaume Oliver Lafont, Sep 20 2009

a(n) = C(n+3,4) + 11*C(n+2,4) + 11*C(n+1,4) + C(n,4). [Worpitzky's identity for powers of 4. See, e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]

a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - Bruno Berselli, May 07 2010

a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - Charlie Marion, Jun 13 2013

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - Ant King, Sep 23 2013

From Amiram Eldar, Jan 20 2021: (Start)

Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).

Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)

Maple

A000583 := n->n^4: seq(A000583(n), n=0..50);
A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); # Zerinvary Lajos, May 25 2008

Mathematica

Range[0, 100]^4 (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)

Prog

(PARI) A000583(n) = n^4 \\ Michael B. Porter, Nov 09 2009
(Haskell)
a000583 = (^ 4)
a000583_list = scanl (+) 0 a005917_list
-- Reinhard Zumkeller, Nov 13 2014, Nov 11 2012
(Maxima) makelist(n^4, n, 0, 30); /* Martin Ettl, Nov 12 2012 */
(Magma) [n^4 : n in [0..50]]; // Wesley Ivan Hurt, Sep 05 2014
(Python)
def a(n): return n**4
print([a(n) for n in range(34)]) # Michael S. Branicky, Nov 10 2022

Crossrefs

Cf. A000538, A005917 (first differences), A000332, A014820, A092181, A092182, A092183.

Cf. A004831, A002646.

Cf. A002593, A260810. - Bruno Berselli, Jul 31 2015

Cf. A002415, A000290, A006008, A132366, A039623, A139584, A071270, A047928, A187756.

Cf. A062392, A231303, A267315.

Sequence in context: A017672 A055013 A080150 * A050751 A014188 A352050

Adjacent sequences: A000580 A000581 A000582 * A000584 A000585 A000586

Keyword

nonn,core,easy,nice,mult

Author

N. J. A. Sloane

Status

approved