A048396 Sums of consecutive noncubes.

0, 27, 315, 1638, 5670, 15345, 35217, 71820, 134028, 233415, 384615, 605682, 918450, 1348893, 1927485, 2689560, 3675672, 4931955, 6510483, 8469630, 10874430, 13796937, 17316585, 21520548, 26504100, 32370975, 39233727, 47214090, 56443338, 67062645, 79223445

Offset: 0

Author

Patrick De Geest, Mar 15 1999

Keywords

• nonn
• easy
• nice

Comments

Relation with triangular numbers: a(n) = 3*((n^3+1) + ((n+1)^3-1)) * A000217(n). Example: a(3) = 3*(first term + last term)*A000217(3) = 3*(28+63)*6 = 1638.

Examples

			Between 3^3 and 4^3 we have: 28 + 29 + ... + 62 + 63 = 1638 = a(3).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000217, A048395, A048397.

Programs

Magma

[(6*n^5+15*n^4+18*n^3+12*n^2+3*n)/2 : n in [0..50]]; // Wesley Ivan Hurt, Apr 10 2015

Maple

A048396:=n->(6*n^5+15*n^4+18*n^3+12*n^2+3*n)/2: seq(A048396(n), n=0..50); # Wesley Ivan Hurt, Apr 10 2015

Mathematica

Table[Total[Range[n^3+1,(n+1)^3-1]], {n,0,30}] (* Harvey P. Dale, Jan 08 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,27,315,1638,5670,15345},40] (* Harvey P. Dale, Nov 02 2024 *)

PARI

a(n)=(6*n^5+15*n^4+18*n^3+12*n^2+3*n)/2 \\ Charles R Greathouse IV, Oct 07 2015

Python

def A048396(n): return n*(n*(n*(n*(6*n + 15) + 18) + 12) + 3)>>1 # Chai Wah Wu, Sep 04 2024

Formula

a(n) = ( 6n^5 + 15n^4 + 18n^3 + 12n^2 + 3n ) / 2.

G.f.: 9*x*(1+x)*(3+14*x+3*x^2)/(1-x)^6. - Colin Barker, Mar 15 2012

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Wesley Ivan Hurt, Apr 10 2015