A048395 Sum of consecutive nonsquares.

0, 5, 26, 75, 164, 305, 510, 791, 1160, 1629, 2210, 2915, 3756, 4745, 5894, 7215, 8720, 10421, 12330, 14459, 16820, 19425, 22286, 25415, 28824, 32525, 36530, 40851, 45500, 50489, 55830, 61535, 67616, 74085, 80954, 88235, 95940, 104081

Offset: 0

Author

Patrick De Geest, Mar 15 1999

Keywords

• nonn
• nice
• easy

Comments

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; ...

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

First differences are in A201279. - J. M. Bergot, Jun 22 2013 [Corrected by Omar E. Pol, Dec 26 2021]

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A048396, A048397, A046092, A001844.

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

Programs

Haskell

a048395 0 = 0
a048395 n = a199771 (2 * n)  -- Reinhard Zumkeller, Oct 26 2015

Mathematica

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 *)

PARI

v0=[1,0,1]; M=[1, 2, 2; -2, -1, -2; 2, 2, 3];
g(v)=v[1]*v[2]*v[3]/(v[1]+v[2]+v[3]);
a(n)=g(v0*M^n);
for(i=0,50,print1(a(i),", ")) \\ Lambert Herrgesell (zero815(AT)googlemail.com), Dec 13 2005

Formula

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

a(n) = Sum_{j=0..n} ((n+j+2)^2 - j^2 + 1). - Zerinvary Lajos, Sep 13 2006

O.g.f.: x(x+5)(1+x)/(1-x)^4. - R. J. Mathar, Jun 12 2008

a(n) = A199771(2*n) for n > 0. - Reinhard Zumkeller, Nov 23 2011

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

E.g.f.: exp(x)*x*(5 + 8*x + 2*x^2). - Stefano Spezia, Jun 25 2022