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).
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
KEYWORD
nonn,nice,easy
AUTHOR
Patrick De Geest, Mar 15 1999
STATUS
approved