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A062392 a(n) = n^4 - (n-1)^4 + (n-2)^4 - ... 0^4. 14
0, 1, 15, 66, 190, 435, 861, 1540, 2556, 4005, 5995, 8646, 12090, 16471, 21945, 28680, 36856, 46665, 58311, 72010, 87990, 106491, 127765, 152076, 179700, 210925, 246051, 285390, 329266, 378015, 431985, 491536, 557040, 628881, 707455 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

Partial sums of A007588. - Jonathan Vos Post, Mar 15 2006

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

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

a(n) is the sum of all positive numbers less than A002378(n). - J. M. Bergot, Aug 30 2013

Except the first term, these are triangular numbers that remain triangular when divided by their index, e.g., 66 divided by 11 gives 6, (see also the Mathematica code). - Waldemar Puszkarz, Sep 14 2017

REFERENCES

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

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000

Milan Janjic, Two Enumerative Functions

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

FORMULA

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).

a(n) = Sum_{i=0..n} A007588(i) for n > 0. - Jonathan Vos Post, Mar 15 2006

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

G.f.: x*(x*(x + 10) + 1)/(1 - x)^5. - Harvey P. Dale, Oct 19 2011

a(n) = A000384(A000217(n)). - Bruno Berselli, Jan 31 2014

a(n) = A110450(n) - A002378(n). - Gionata Neri, May 13 2015

EXAMPLE

From Bruno Berselli, Oct 30 2017: (Start)

After 0:

1   =                 -(1) + (2);

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

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

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

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

MAPLE

a := n -> (2*n^2+n^3-1)*n/2; # Peter Luschny, Jul 12 2009

MATHEMATICA

Table[n (n + 1) (n^2 + n - 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 15, 66, 190}, 40] (* Harvey P. Dale, Oct 19 2011 *)

lst=Select[Range[0, 2*10^3], OddQ@Sqrt[4#+5]&]//#(#+1)/2&; PrependTo[lst, 0]; lst (* Waldemar Puszkarz, Sep 14 2017 *)

PROG

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

CROSSREFS

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.

Cf. A000217, A000384, A007588.

Cf. A236770 (see crossrefs).

Sequence in context: A284898 A033653 A088058 * A211787 A265141 A211917

Adjacent sequences:  A062389 A062390 A062391 * A062393 A062394 A062395

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jun 21 2001

STATUS

approved

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Last modified August 17 22:32 EDT 2018. Contains 313817 sequences. (Running on oeis4.)