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

1,2

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*(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, 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).

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

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 *)

Prog

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

Crossrefs

Cf. A002417, A007585, A076455-A076459, A233329.

Sequence in context: A356738 A044272 A044653 * A208538 A303681 A304132

Adjacent sequences: A076451 A076452 A076453 * A076455 A076456 A076457

Keyword

nonn,easy

Author

Floor van Lamoen, Oct 13 2002

Extensions

Comments rewritten from Bruno Berselli, Mar 01 2012

More terms from Vincenzo Librandi, Dec 30 2013

Status

approved