A065113 Sum of the squares of the a(n)-th and the (a(n)+1)st triangular numbers (A000217) is a perfect square.

6, 40, 238, 1392, 8118, 47320, 275806, 1607520, 9369318, 54608392, 318281038, 1855077840, 10812186006, 63018038200, 367296043198, 2140758220992, 12477253282758, 72722761475560, 423859315570606, 2470433131948080, 14398739476117878, 83922003724759192

Offset

1,1

Comments

The sequence of square roots of the sum of the squares of the n-th and the (n+1)st triangular numbers is A046176.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

M. Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT] (2008)

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Formula

a(n) = 2*A001652(n) = -1 + A002315(n).

a(n) - a(n-1) = A003499(n).

From Michael Somos, Apr 07 2003: (Start)

G.f.: 2*x*(3-x)/((1-6*x+x^2)*(1-x)).

a(n) = 6*a(n-1) - a(n-2) + 4.

a(-1-n) = -a(n) - 2. (End)

a(1)=6, a(2)=40, a(3)=238, a(n) = 7*a(n-1)-7*a(n-2)+a(n-3). - Harvey P. Dale, Dec 27 2011

a(n)^2 + (a(n)+2)^2 = A075870(n+1)^2 = A165518(n+1). - Joerg Arndt, Feb 15 2012

a(n) = (-2-(3-2*sqrt(2))^n*(-1+sqrt(2))+(1+sqrt(2))*(3+2*sqrt(2))^n)/2. - Colin Barker, Mar 05 2016

From Klaus Purath, Sep 05 2021: (Start)

(a(n+1) - a(n) - a(n-1) + a(n-2))/8 = A005319(n), for n >= 3.

((a(n) - a(n-1))^2)/2 - 2 = A005319(n)^2 = 2*A132592(n), for n>= 2.

a(n) = A265278(2*n+1).

a(n) = A293004(2*n+1).

a(n) = A213667(2*n).

a(n) = Sum_{k=1..n} A003499(k). (End)

Example

T6 = 21 and T7 = 28, 21^2 + 28^2 = 441 + 784 = 1225 = 35^2.

Mathematica

CoefficientList[ Series[2*(x - 3)/(-1 + 7x - 7x^2 + x^3), {x, 0, 24} ], x]
LinearRecurrence[{7, -7, 1}, {6, 40, 238}, 41] (* Harvey P. Dale, Dec 27 2011 *)

Prog

(PARI) a(n)=-1+subst(poltchebi(abs(n+1))-poltchebi(abs(n)), x, 3)/2
(PARI) Vec(2*x*(3-x)/((1-6*x+x^2)*(1-x)) + O(x^40)) \\ Colin Barker, Mar 05 2016

Crossrefs

Cf. A001652, A002315, A003499 (first differences), A065651.

Sequence in context: A254945 A026077 A348601 * A052518 A135032 A122074

Adjacent sequences: A065110 A065111 A065112 * A065114 A065115 A065116

Keyword

nonn,easy

Author

Robert G. Wilson v, Nov 12 2001

Status

approved