A165518 Perfect squares (A000290) that can be expressed as the sum of four consecutive triangular numbers (A000217).

4, 100, 3364, 114244, 3880900, 131836324, 4478554084, 152139002500, 5168247530884, 175568277047524, 5964153172084900, 202605639573839044, 6882627592338442564, 233806732499933208100, 7942546277405390632804, 269812766699283348307204, 9165691521498228451812100, 311363698964240484013304164

Offset

1,1

Comments

As T(n) + T(n+1) = (n+1)^2 and T(n+2) + T(n+3) = (n+3)^2, it follows that the equation T(n) + T(n+1) + T(n+2) + T(n+3) = s^2 becomes (n+1)^2 + (n+3)^2 = s^2. Hence the solutions to this equation correspond to those Pythagorean triples with shorter legs that differ by two, such as 6^2 + 8^2 = 10^2.

Terms are the squares of the hypotenuses of Pythagorean triangles where other two sides are m and m+2, excepting the initial 4. See A075870. - Richard R. Forberg, Aug 15 2013

Links

Harvey P. Dale, Table of n, a(n) for n = 1..600

Tom Beldon and Tony Gardiner, Triangular Numbers and Perfect Squares, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Formula

a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3).

a(n) = 34*a(n-1) - a(n-2) - 32.

a(n) = (2 + (3+2*sqrt(2))^(2*n+1) + (3-2*sqrt(2))^(2*n+1))/2.

a(n) = ceiling((1/2)*(2 + (3+2*sqrt(2))^(2n+1))).

G.f.: 4*x*(x^2-10*x+1)/((1-x)*(x^2-34*x+1)).

a(n) = 4*A008844(n-1). - R. J. Mathar, Dec 14 2010

a(n) = A075870(n)^2. - Richard R. Forberg, Aug 15 2013

Example

As the third perfect square that can be expressed as the sum of four consecutive triangular numbers is 3364 = T(39) + T(40) + T(41) + T(42), we have a(3)=3364.
The first term, 4, equals T(-1) + T(0) + T(1) + T(2).

Maple

A165518:=n->(1/2)*(2+(3+2*sqrt(2))^(2*n+1)+(3-2*sqrt(2))^(2*n+1)); seq(A165518(k), k=1..20); # Wesley Ivan Hurt, Oct 24 2013

Mathematica

TriangularNumber[n_]:=1/2 n (n+1); data=Select[Range[10^7], IntegerQ[Sqrt[ TriangularNumber[ # ]+TriangularNumber[ #+1]+TriangularNumber[ #+2]+TriangularNumber[ #+3]]] &]; 2(#^2+4#+5)&/@data
t={4, 100}; Do[AppendTo[t, 34 t[[-1]] - t[[-2]] - 32], {20}]; t
LinearRecurrence[{35, -35, 1}, {4, 100, 3364}, 20] (* Harvey P. Dale, May 22 2012 *)

Prog

(PARI) x='x+O('x^50); Vec(4*x*(1-10*x+x^2)/((1-x)*(1-34*x+x^2))) \\ G. C. Greubel, Oct 21 2018
(Magma) I:=[4, 100, 3364]; [n le 3 select I[n] else 35*Self(n-1) - 35*Self(n-2) +Self(n-3): n in [1..50]]; // G. C. Greubel, Oct 21 2018

Crossrefs

Cf. A000290, A000217, A165516 (squares that can be expressed as the sum of three consecutive triangular numbers), A029549, A075870.

Sequence in context: A244352 A173987 A052144 * A127776 A266524 A224167

Adjacent sequences: A165515 A165516 A165517 * A165519 A165520 A165521

Keyword

easy,nice,nonn

Author

Ant King, Sep 28 2009

Extensions

Extended by T. D. Noe, Dec 09 2010

Status

approved