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A076454
Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.
9
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
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
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
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