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A050410 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2.

%I A050410 #36 Feb 16 2022 23:58:34
%S A050410 0,1,13,50,126,255,451,728,1100,1581,2185,2926,3818,4875,6111,7540,
%T A050410 9176,11033,13125,15466,18070,20951,24123,27600,31396,35525,40001,
%U A050410 44838,50050,55651,61655,68076,74928,82225,89981,98210,106926,116143
%N A050410 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2.
%C A050410 Starting with offset 1 = binomial transform of [1, 12, 25, 14, 0, 0, 0, ...]. - _Gary W. Adamson_, Jan 09 2009
%H A050410 Vincenzo Librandi, <a href="/A050410/b050410.txt">Table of n, a(n) for n = 0..1000</a>
%H A050410 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A050410 a(n) = n*(7*n-1)*(2*n-1)/6.
%F A050410 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=13, a(3)=50. - _Harvey P. Dale_, Feb 29 2012
%F A050410 G.f.: x*(1 + 9*x + 4*x^2)/(1-x)^4. - _Colin Barker_, Mar 23 2012
%F A050410 E.g.f.: x*(6 + 33*x + 14*x^2)*exp(x)/6. - _G. C. Greubel_, Oct 30 2019
%e A050410 1^2 + 1;
%e A050410 2^2 + 3^2 = 13;
%e A050410 3^2 + 4^2 + 5^2 = 50; ...
%p A050410 seq(n*(7*n-1)*(2*n-1)/6, n=0..36); # _Zerinvary Lajos_, Dec 01 2006
%t A050410 Table[Sum[k^2,{k,n,2n-1}],{n,0,40}] (* or *) LinearRecurrence[{4,-6,4, -1}, {0,1,13,50},40] (* _Harvey P. Dale_, Feb 29 2012 *)
%o A050410 (PARI) for(n=1,100,print1(sum(i=0,n-1,(n+i)^2),","))
%o A050410 (PARI) vector(40, n, (n-1)*(7*n-8)*(2*n-3)/6) \\ _G. C. Greubel_, Oct 30 2019
%o A050410 (Magma) [n*(7*n-1)*(2*n-1)/6: n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2012
%o A050410 (Sage) [n*(7*n-1)*(2*n-1)/6 for n in (0..40)] # _G. C. Greubel_, Oct 30 2019
%o A050410 (GAP) List([0..40], n-> n*(7*n-1)*(2*n-1)/6); # _G. C. Greubel_, Oct 30 2019
%Y A050410 Cf. A072474, A240137.
%K A050410 nonn,easy,nice
%O A050410 0,3
%A A050410 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999