A076454 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.

1, 21, 102, 310, 735, 1491, 2716, 4572, 7245, 10945, 15906, 22386, 30667, 41055, 53880, 69496, 88281, 110637, 136990, 167790, 203511, 244651, 291732, 345300, 405925, 474201, 550746, 636202, 731235, 836535, 952816, 1080816, 1221297, 1375045, 1542870, 1725606, 1924111

Offset: 1

Author

Floor van Lamoen, Oct 13 2002

Keywords

• nonn
• easy

Comments

This sequence is related to A007585 by a(n) = n*A007585(n) - Sum_{i=0..n-1} A007585(i). - Vincenzo Librandi, Aug 08 2010

In fact, this is the case d=4 in the identity n*(n*(n+1)*(2*d*n-2*d+3)/6) - Sum_{k=0..n-1} k*(k+1)*(2*d*k-2*d+3)/6 = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12. - Bruno Berselli, Mar 01 2012

Bisection of A233329 (up to an offset). - L. Edson Jeffery, Jan 23 2014

References

• Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..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. A002417, A007585, A076455-A076459, A233329.

Programs

Magma

[n*(n+1)*(2*n^2-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

Maple

seq(1/2*n*(n+1)*(2*n^2-1),n=1..40);

Mathematica

CoefficientList[Series[(1 + 16 x + 7 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,21,102,310,735},40] (* Harvey P. Dale, Jun 30 2023 *)

Formula

a(n) = n*(n+1)*(2*n^2-1)/2.

G.f.: x*(1+16*x+7*x^2)/(1-x)^5.

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n>=6, with a(1)=1, a(2)=21, a(3)=102, a(4)=310, a(5)=735. - L. Edson Jeffery, Dec 30 2013

Extensions

Comments rewritten from Bruno Berselli, Mar 01 2012

More terms from Vincenzo Librandi, Dec 30 2013