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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.

%I A076454 #35 Aug 18 2025 23:36:29
%S A076454 1,21,102,310,735,1491,2716,4572,7245,10945,15906,22386,30667,41055,
%T A076454 53880,69496,88281,110637,136990,167790,203511,244651,291732,345300,
%U A076454 405925,474201,550746,636202,731235,836535,952816,1080816,1221297,1375045,1542870,1725606,1924111
%N A076454 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.
%C A076454 This sequence is related to A007585 by a(n) = n*A007585(n) - Sum_{i=0..n-1} A007585(i). - _Vincenzo Librandi_, Aug 08 2010
%C A076454 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
%C A076454 Bisection of A233329 (up to an offset). - _L. Edson Jeffery_, Jan 23 2014
%D A076454 Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.
%H A076454 Vincenzo Librandi, <a href="/A076454/b076454.txt">Table of n, a(n) for n = 1..1000</a>
%H A076454 B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H A076454 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A076454 a(n) = n*(n+1)*(2*n^2-1)/2.
%F A076454 G.f.: x*(1+16*x+7*x^2)/(1-x)^5.
%F A076454 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
%p A076454 seq(1/2*n*(n+1)*(2*n^2-1),n=1..40);
%t A076454 CoefficientList[Series[(1 + 16 x + 7 x^2)/(1 - x)^5, {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 30 2013 *)
%t A076454 LinearRecurrence[{5,-10,10,-5,1},{1,21,102,310,735},40] (* _Harvey P. Dale_, Jun 30 2023 *)
%o A076454 (Magma) [n*(n+1)*(2*n^2-1)/2: n in [1..50]]; // _Vincenzo Librandi_, Dec 30 2013
%Y A076454 Cf. A002417, A007585, A076455-A076459, A233329.
%K A076454 nonn,easy
%O A076454 1,2
%A A076454 _Floor van Lamoen_, Oct 13 2002
%E A076454 Comments rewritten from _Bruno Berselli_, Mar 01 2012
%E A076454 More terms from _Vincenzo Librandi_, Dec 30 2013