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%I M5004 N2154 #247 Nov 13 2023 07:57:12
%S 0,1,16,81,256,625,1296,2401,4096,6561,10000,14641,20736,28561,38416,
%T 50625,65536,83521,104976,130321,160000,194481,234256,279841,331776,
%U 390625,456976,531441,614656,707281,810000,923521,1048576,1185921
%N Fourth powers: a(n) = n^4.
%C 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
%C Totally multiplicative sequence with a(p) = p^4 for prime p. - _Jaroslav Krizek_, Nov 01 2009
%C 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
%C 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
%C 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
%C Does not satisfy Benford's law [Ross, 2012]. - _N. J. A. Sloane_, Feb 08 2017
%C 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
%D 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).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000583/b000583.txt">Table of n, a(n) for n = 0..1000</a>
%H Henry Bottomley, <a href="/A000583/a000583.gif">Illustration of initial terms</a>
%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>
%H Ralph Greenberg, <a href="http://www.math.washington.edu/~greenber/MathPoet.html">Math for Poets</a>.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H Sameen Ahmed Khan, <a href="https://doi.org/10.12732/ijam.v33i2.6">Sums of the powers of reciprocals of polygonal numbers</a>, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Kenneth A. Ross, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.1.036">First Digits of Squares and Cubes</a>, Math. Mag. 85 (2012) 36-42.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(n) = A123865(n)+1 = A002523(n)-1.
%F Multiplicative with a(p^e) = p^(4e). - _David W. Wilson_, Aug 01 2001
%F 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).
%F Dirichlet generating function: zeta(s-4). - _Franklin T. Adams-Watters_, Sep 11 2005
%F 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
%F Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - _Jaume Oliver Lafont_, Sep 20 2009
%F 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]
%F a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - _Bruno Berselli_, May 07 2010
%F a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - _Charlie Marion_, Jun 13 2013
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - _Ant King_, Sep 23 2013
%F From _Amiram Eldar_, Jan 20 2021: (Start)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).
%F Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)
%p A000583 := n->n^4: seq(A000583(n), n=0..50);
%p A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation; gives sequence without initial zero
%p with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); # _Zerinvary Lajos_, May 25 2008
%t Range[0,100]^4 (* _Vladimir Joseph Stephan Orlovsky_, Mar 14 2011 *)
%o (PARI) A000583(n) = n^4 \\ _Michael B. Porter_, Nov 09 2009
%o (Haskell)
%o a000583 = (^ 4)
%o a000583_list = scanl (+) 0 a005917_list
%o -- _Reinhard Zumkeller_, Nov 13 2014, Nov 11 2012
%o (Maxima) makelist(n^4,n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o (Magma) [n^4 : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 05 2014
%o (Python)
%o def a(n): return n**4
%o print([a(n) for n in range(34)]) # _Michael S. Branicky_, Nov 10 2022
%Y Cf. A000538, A005917 (first differences), A000332, A014820, A092181, A092182, A092183.
%Y Cf. A004831, A002646.
%Y Cf. A002593, A260810. - _Bruno Berselli_, Jul 31 2015
%Y Cf. A002415, A000290, A006008, A132366, A039623, A139584, A071270, A047928, A187756.
%Y Cf. A062392, A231303, A267315.
%K nonn,core,easy,nice,mult
%O 0,3
%A _N. J. A. Sloane_
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