A135405 Sequence where the sum of each pair of consecutive elements is a square.

0, 1, 8, 8, 17, 19, 30, 34, 47, 53, 68, 76, 93, 103, 122, 134, 155, 169, 192, 208, 233, 251, 278, 298, 327, 349, 380, 404, 437, 463, 498, 526, 563, 593, 632, 664, 705, 739, 782, 818, 863, 901, 948, 988, 1037, 1079, 1130, 1174, 1227, 1273, 1328, 1376, 1433, 1483

Offset

0,3

Comments

This covers squares of all consecutively increasing integers with the exception of 2.

It is actually possible to cover all nonnegative integers by using the given formula starting with n=-2, thus giving terms 2, -2, 3, 1, 8, 8, 17, 19, 30, etc. - Vladimir Joseph Stephan Orlovsky, Feb 12 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

a(n) = (n+2)*(n+1)/2 + 2*(-1)^n for n>0.

From R. J. Mathar, Dec 12 2007: (Start)

O.g.f.: x*(1 +6*x -8*x^2 +3*x^3)/((1-x)^3*(1+x)) = -3 +1/(1-x)^3 + 2/(1+x).

a(n) = A000217(n+1) + 2*(-1)^n if n>0.

(End)

E.g.f.: -3 + 2*exp(-x) + (1/2)*(2 + 4*x + x^2)*exp(x). - G. C. Greubel, Oct 12 2016

From Colin Barker, Oct 13 2016: (Start)

a(n) = (-4*(-1)^n+n+n^2)/2 for n>1.

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4.

(End)

Example

a(1) = 1 because 0 + 1 = 1^2.
a(2) = 8 because 1 + 8 = 9 = 3^2.
a(3) = 8 because 8 + 8 = 16 = 4^2.

Mathematica

a=1; lst={0, a}; Do[a=n^2-a; AppendTo[lst, a], {n, 3, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
Table[(n+2)*(n+1)/2 + 2*(-1)^n, {n, 0, 25}] (* G. C. Greubel, Oct 12 2016 *)

Prog

(Magma) [0] cat [(n+2)*(n+1)/2+2*(-1)^n: n in [1..60]]; // Vincenzo Librandi, Feb 14 2015
(PARI) concat(0, Vec(x*(1+6*x-8*x^2+3*x^3)/((1-x)^3*(1+x)) + O(x^60))) \\ Colin Barker, Oct 13 2016

Crossrefs

Sequence in context: A171188 A145909 A168409 * A006784 A214830 A168456

Adjacent sequences: A135402 A135403 A135404 * A135406 A135407 A135408

Keyword

nonn,easy

Author

Alexander R. Povolotsky, Dec 11 2007, Apr 02 2008

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Dec 17 2008

Status

approved