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A165518 Perfect squares (A000290) that can be expressed as the sum of four consecutive triangular numbers (A000217). 4
4, 100, 3364, 114244, 3880900, 131836324, 4478554084, 152139002500, 5168247530884, 175568277047524, 5964153172084900, 202605639573839044, 6882627592338442564, 233806732499933208100, 7942546277405390632804, 269812766699283348307204, 9165691521498228451812100, 311363698964240484013304164 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 16 12:19 EDT 2019. Contains 325079 sequences. (Running on oeis4.)