A048397 -id:A048397 - cp's OEIS frontend

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

Patrick De Geest, Mar 15 1999

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]

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

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

Cf. A048396, A048397, A046092, A001844.

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

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

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 *)

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

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

A048396 Sums of consecutive noncubes.

Original entry on oeis.org

0, 27, 315, 1638, 5670, 15345, 35217, 71820, 134028, 233415, 384615, 605682, 918450, 1348893, 1927485, 2689560, 3675672, 4931955, 6510483, 8469630, 10874430, 13796937, 17316585, 21520548, 26504100, 32370975, 39233727, 47214090, 56443338, 67062645, 79223445

Offset: 0

Patrick De Geest, Mar 15 1999

Relation with triangular numbers: a(n) = 3*((n^3+1) + ((n+1)^3-1)) * A000217(n). Example: a(3) = 3*(first term + last term)*A000217(3) = 3*(28+63)*6 = 1638.

			Between 3^3 and 4^3 we have: 28 + 29 + ... + 62 + 63 = 1638 = a(3).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A048395, A048397.

[(6*n^5+15*n^4+18*n^3+12*n^2+3*n)/2 : n in [0..50]]; // Wesley Ivan Hurt, Apr 10 2015

A048396:=n->(6*n^5+15*n^4+18*n^3+12*n^2+3*n)/2: seq(A048396(n), n=0..50); # Wesley Ivan Hurt, Apr 10 2015

Table[Total[Range[n^3+1,(n+1)^3-1]], {n,0,30}] (* Harvey P. Dale, Jan 08 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,27,315,1638,5670,15345},40] (* Harvey P. Dale, Nov 02 2024 *)

a(n)=(6*n^5+15*n^4+18*n^3+12*n^2+3*n)/2 \\ Charles R Greathouse IV, Oct 07 2015

def A048396(n): return n*(n*(n*(n*(6*n + 15) + 18) + 12) + 3)>>1 # Chai Wah Wu, Sep 04 2024

a(n) = ( 6n^5 + 15n^4 + 18n^3 + 12n^2 + 3n ) / 2.
G.f.: 9*x*(1+x)*(3+14*x+3*x^2)/(1-x)^6. - Colin Barker, Mar 15 2012
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Wesley Ivan Hurt, Apr 10 2015

cp's OEIS Frontend

A048395 Sum of consecutive nonsquares.

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