A172045 a(n) = (9*n^4+10*n^3-3*n^2-4*n)/12.

0, 1, 17, 80, 240, 565, 1141, 2072, 3480, 5505, 8305, 12056, 16952, 23205, 31045, 40720, 52496, 66657, 83505, 103360, 126560, 153461, 184437, 219880, 260200, 305825, 357201, 414792, 479080, 550565, 629765, 717216, 813472, 919105, 1034705

Offset: 0

Author

Vincenzo Librandi, Jan 24 2010

Keywords

• nonn
• easy

Comments

The sequence is related to A002414 (octagonal pyramidal numbers) by a(n) = n*A002414(n)-sum(A002414(i), i=1..n-1) for n>0.

This is the case d=3 in the identity n*(n*(n+1)*(2*d*n-2*d+3)/6)-sum(k*(k+1)*(2*d*k-2*d+3)/6, k=0..n-1) = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12. - Bruno Berselli, Nov 03 2010

Also, the sequence is related to A000567 by a(n) = sum( i*A000567(i), i=0..n ). [Bruno Berselli, Dec 19 2013]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000567, A002414.

Programs

Magma

[(9*n^4+10*n^3-3*n^2-4*n)/12: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

Mathematica

CoefficientList[Series[x (1 + 12 x + 5 x^2)/(1 - x)^5,{x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,17,80,240},40] (* Harvey P. Dale, Aug 25 2019 *)

Formula

a(n) = n*(n+1)*(9*n^2+n-4)/12. - Bruno Berselli, Apr 21 2010

G.f. -x*(1 +12*x +5*x^2) / (x - 1)^5 . - R. J. Mathar, Nov 17 2011

Extensions

Edited by Bruno Berselli, Oct 06 - 12 2010