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A050410 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2. 8

%I

%S 0,1,13,50,126,255,451,728,1100,1581,2185,2926,3818,4875,6111,7540,

%T 9176,11033,13125,15466,18070,20951,24123,27600,31396,35525,40001,

%U 44838,50050,55651,61655,68076,74928,82225,89981,98210,106926,116143

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

%C Starting with offset 1 = binomial transform of [1, 12, 25, 14, 0, 0, 0,...]. [From _Gary W. Adamson_, Jan 09 2009]

%H Vincenzo Librandi, <a href="/A050410/b050410.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n)=n*(7*n-1)*(2*n-1)/6.

%F a(0)=0, a(1)=1, a(2)=13, a(3)=50, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). [_Harvey P. Dale_, Feb 29 2012]

%F G.f.: x*(1+9*x+4*x^2)/(1-x)^4. [_Colin Barker_, Mar 23 2012]

%e 1^2 + 1; 2^2 + 3^2 = 13; 3^2 + 4^2 + 5^2 = 50; ...

%p seq(add((n+k+1)^2,k=0..n),n=-1..36); - _Zerinvary Lajos_, Dec 01 2006

%t 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 (PARI) for(n=1,100,print1(sum(i=0,n-1,(n+i)^2),","))

%o (MAGMA) [n*(7*n-1)*(2*n-1)/6: n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2012

%Y Cf. A072474, A240137.

%K nonn,easy,nice

%O 0,3

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

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Last modified October 18 18:10 EDT 2018. Contains 316323 sequences. (Running on oeis4.)