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A076465 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.

%I A076465 #20 Jul 05 2025 08:41:15
%S A076465 1,571,12938,115270,630755,2543401,8307796,23249388,57792165,
%T A076465 130790935,274285726,540036146,1008233863,1798831685,3085968040,
%U A076465 5116005976,8229746121,12889413363,19711057330,29503047070,43311380651,62472570721,88674907388,124028940100
%N A076465 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.
%D A076465 Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.
%H A076465 Vincenzo Librandi, <a href="/A076465/b076465.txt">Table of n, a(n) for n = 1..1000</a>
%H A076465 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A076465 a(n) = n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1)/6.
%F A076465 G.f.: x*(1+562*x+7835*x^2+19300*x^3+11255*x^4+1354*x^5+13*x^6)/(1-x)^9.
%F A076465 a(1)=1, a(2)=571, a(3)=12938, a(4)=115270, a(5)=630755, a(6)=2543401, a(7)=8307796, a(8)=23249388, a(9)=57792165, a(n)=9*a(n-1)- 36*a(n-2)+ 84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - _Harvey P. Dale_, Sep 05 2015
%p A076465 seq(1/6*n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1),n=1..25);
%t A076465 CoefficientList[Series[(1 + 562 x + 7835 x^2 + 19300 x^3 + 11255 x^4 + 1354 x^5 + 13 x^6)/(1 - x)^9, {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 30 2013 *)
%t A076465 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,571,12938,115270,630755,2543401,8307796,23249388,57792165},30] (* _Harvey P. Dale_, Sep 05 2015 *)
%o A076465 (Magma) [n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1)/6: n in [1..50]]; // _Vincenzo Librandi_, Dec 30 2013
%Y A076465 Cf. A076389, A076459-A076464.
%K A076465 easy,nonn
%O A076465 1,2
%A A076465 _Floor van Lamoen_, Oct 13 2002
%E A076465 More terms from _Vincenzo Librandi_, Dec 30 2013