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A048395 Sum of consecutive nonsquares. 14

%I #58 Jun 30 2022 05:59:03

%S 0,5,26,75,164,305,510,791,1160,1629,2210,2915,3756,4745,5894,7215,

%T 8720,10421,12330,14459,16820,19425,22286,25415,28824,32525,36530,

%U 40851,45500,50489,55830,61535,67616,74085,80954,88235,95940,104081

%N Sum of consecutive nonsquares.

%C Relationship with natural numbers: a(4) = (first term + last term)*n = (10+15)*3 = (25)*3 = 75; a(5) = (17+24)*4 = (41)*4 = 164; ...

%C Also (X*Y*Z)/(X+Y+Z) of primitive Pythagorean triples (X,Y,Z=Y+1) as described in A046092 and A001844. - Lambert Herrgesell (zero815(AT)googlemail.com), Dec 13 2005

%C First differences are in A201279. - _J. M. Bergot_, Jun 22 2013 [Corrected by _Omar E. Pol_, Dec 26 2021]

%H Vincenzo Librandi, <a href="/A048395/b048395.txt">Table of n, a(n) for n = 0..10000</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.

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

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F a(n) = 2*n^3 + 2*n^2 + n.

%F a(n) = Sum_{j=0..n} ((n+j+2)^2 - j^2 + 1). - _Zerinvary Lajos_, Sep 13 2006

%F O.g.f.: x(x+5)(1+x)/(1-x)^4. - _R. J. Mathar_, Jun 12 2008

%F a(n) = A199771(2*n) for n > 0. - _Reinhard Zumkeller_, Nov 23 2011

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=5, a(2)=26, a(3)=75. - _Harvey P. Dale_, Nov 01 2013

%F E.g.f.: exp(x)*x*(5 + 8*x + 2*x^2). - _Stefano Spezia_, Jun 25 2022

%e Between 3^2 and 4^2 we have 10+11+12+13+14+15 which is 75 or a(4).

%t Table[n(1+2*n(1+n)),{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,5,26,75},40] (* _Harvey P. Dale_, Nov 01 2013 *)

%o (PARI) v0=[1,0,1]; M=[1, 2, 2; -2, -1, -2; 2, 2, 3];

%o g(v)=v[1]*v[2]*v[3]/(v[1]+v[2]+v[3]);

%o a(n)=g(v0*M^n);

%o for(i=0,50,print1(a(i),", ")) \\ Lambert Herrgesell (zero815(AT)googlemail.com), Dec 13 2005

%o (Haskell)

%o a048395 0 = 0

%o a048395 n = a199771 (2 * n) -- _Reinhard Zumkeller_, Oct 26 2015

%Y Cf. A048396, A048397, A046092, A001844.

%Y Cf. A001844, A046092, A086849, A199771, A201279.

%K nonn,nice,easy

%O 0,2

%A _Patrick De Geest_, Mar 15 1999

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