A062392 - OEIS

%I #72 Aug 03 2024 19:26:21

%S 0,1,15,66,190,435,861,1540,2556,4005,5995,8646,12090,16471,21945,

%T 28680,36856,46665,58311,72010,87990,106491,127765,152076,179700,

%U 210925,246051,285390,329266,378015,431985,491536,557040,628881,707455,793170,886446,987715

%N a(n) = n^4 - (n-1)^4 + (n-2)^4 - ... 0^4.

%C Number of edges in the join of two complete graphs of order n^2 and n, K_n^2 * K_n - _Roberto E. Martinez II_, Jan 07 2002

%C The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus a(k) = |2^(-5)(P(4,1)-(-1)^k P(4,2k+1))|. - _Peter Luschny_, Jul 12 2009

%C Define an infinite symmetric array by T(n,m) = n*(n-1) + m for 0 <= m <= n and T(n,m) = T(m,n), n >= 0. Then a(n) is the sum of terms in the top left (n+1) X (n+1) subarray: a(n) = Sum_{r=0..n} Sum_{c=0..n} T(r,c). - _J. M. Bergot_, Jul 05 2013

%C a(n) is the sum of all positive numbers less than A002378(n). - _J. M. Bergot_, Aug 30 2013

%C Except the first term, these are triangular numbers that remain triangular when divided by their index, e.g., 66 divided by 11 gives 6. - _Waldemar Puszkarz_, Sep 14 2017

%D T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

%H Harry J. Smith, <a href="/A062392/b062392.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n*(n+1)*(n^2 + n - 1)/2 = n^4 - a(n-1) = A000583(n) - a(n) = A000217(A028387(n-1)) = A000217(n)*A028387(n-1).

%F a(n) = Sum_{i=0..n} A007588(i) for n > 0. - _Jonathan Vos Post_, Mar 15 2006

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - _Harvey P. Dale_, Oct 19 2011

%F G.f.: x*(x*(x + 10) + 1)/(1 - x)^5. - _Harvey P. Dale_, Oct 19 2011

%F a(n) = A000384(A000217(n)). - _Bruno Berselli_, Jan 31 2014

%F a(n) = A110450(n) - A002378(n). - _Gionata Neri_, May 13 2015

%F Sum_{n>=1} 1/a(n) = tan(sqrt(5)*Pi/2)*2*Pi/sqrt(5). - _Amiram Eldar_, Jan 22 2024

%F a(n) = sqrt(144*A288876(n-2) + 72*A006542(n+2) + A000537(n)). - _Yasser Arath Chavez Reyes_, Jul 22 2024

%e From _Bruno Berselli_, Oct 30 2017: (Start)

%e After 0:

%e 1   =                 -(1) + (2);

%e 15  =             -(1 + 2) + (3 + 4 + 5 + 2*3);

%e 66  =         -(1 + 2 + 3) + (4 + 5 + 6 + 7 + ... + 11 + 3*4);

%e 190 =     -(1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + ... + 19 + 4*5);

%e 435 = -(1 + 2 + 3 + 4 + 5) + (6 + 7 + 8 + 9 + ... + 29 + 5*6), etc. (End)

%p a := n -> (2*n^2+n^3-1)*n/2; # _Peter Luschny_, Jul 12 2009

%t Table[n (n + 1) (n^2 + n - 1)/2, {n, 0, 40}] (* _Harvey P. Dale_, Oct 19 2011 *)

%o (PARI) { a=0; for (n=0, 1000, write("b062392.txt", n, " ", a=n^4 - a) ) } \\ _Harry J. Smith_, Aug 07 2009

%Y Cf. A000538, A000583. A062393 provides the result for 5th powers, A011934 for cubes, A000217 for squares, A001057 (unsigned) for nonnegative integers, A000035 (offset) for 0th powers.

%Y Cf. A000217, A000384, A007588.

%Y Cf. A236770 (see crossrefs).

%K nonn,easy

%O 0,3

%A _Henry Bottomley_, Jun 21 2001