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

0,2

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]

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A048396, A048397, A046092, A001844.

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

Sequence in context: A273833 A273849 A273781 * A309451 A081886 A081530

Adjacent sequences: A048392 A048393 A048394 * A048396 A048397 A048398

Keyword

nonn,nice,easy

Author

Patrick De Geest, Mar 15 1999

Status

approved