OFFSET
1,3
COMMENTS
Polynexus numbers of order 6.
A polynexus (subtractive) function is composed of two or more subtracted nexus numbers divided by an integer x. The general form of the formula is a(n)=((n^p-(n-1)^p)-(n^q-(n-1)^q))/x, where n, p, q and x are integers.
Already known: ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24, giving A006322 for n>1; ((n^4-(n-1)^4) - (n^2-(n-1)^2))/12, giving A000330; ((n^3-(n-1)^3) - (n^1-(n-1)^1))/6, giving A000217; ((n^2-(n-1)^2) - (n^1-(n-1)^1))/2, giving n; ((n^2-(n-1)^2) - (n^0-(n-1)^0))/1, giving 2*n-1. In those examples, x is equal to 1,2,6,12,24, and 3 is also possible.
Also number of monotone n-weightings of complete bipartite digraph K(3,2) if offset were 0; cf. A085464-A085465. - Goran Kilibarda, Vladeta Jovovic, Jul 01 2003
Partial sums of A037270. - J. M. Bergot, Jun 07 2012
LINKS
Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n+1) = Sum_{i=1..n} (i^2 + i^4)/2 = n*(2*n+1)*(n+1)*(3*n^2+3*n+4)/60. - Vladeta Jovovic, Mar 17 2006
G.f.: x^2*(x+1)*(1+4*x+x^2)/(1-x)^6. - Bruno Berselli, Feb 13 2012
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^3. - Enrique Pérez Herrero, Jan 16 2013
E.g.f.: x^2*(30 + 80*x + 45*x^2 + 6*x^3)*exp(x)/60. - G. C. Greubel, Jun 19 2019
MATHEMATICA
Table[((n^6 -(n-1)^6) - (n^2 -(n-1)^2))/60, {n, 30}] (* Bruno Berselli, Feb 13 2012 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 11, 56, 192, 517}, 30] (* Harvey P. Dale, Feb 21 2023 *)
PROG
(PARI) a(n) = n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 \\ Charles R Greathouse IV, Jan 16 2013
(Magma) [n*(6*n^4-15*n^3+20*n^2-15*n+4)/60: n in [1..30]]; // G. C. Greubel, Jun 19 2019
(Sage) [n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 for n in (1..30)] # G. C. Greubel, Jun 19 2019
(GAP) List([1..30], n-> n*(6*n^4-15*n^3+20*n^2-15*n+4)/60) # G. C. Greubel, Jun 19 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Xavier Acloque, Jan 22 2003
STATUS
approved