%I #50 Feb 16 2024 06:37:53

%S 0,3,8,35,84,351,836,3479,8280,34443,81968,340955,811404,3375111,

%T 8032076,33410159,79509360,330726483,787061528,3273854675,7791105924,

%U 32407820271,77123997716,320804348039,763448871240,3175635660123,7557364714688,31435552253195

%N Second member of Diophantine pair (m,k) that satisfies 6*(m^2 + m) = k^2 + k: a(n) = k.

%C The corresponding m are given in A077288.

%H Colin Barker, <a href="/A077291/b077291.txt">Table of n, a(n) for n = 0..1000</a>

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2101.00998">Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers</a>, arXiv:2101.00998 [math.NT], 2021.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.13494">Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations</a>, arXiv:2102.13494 [math.NT], 2021.

%H Vladimir Pletser, <a href="https://www.researchgate.net/profile/Vladimir-Pletser/publication/359808848_USING_PELL_EQUATION_SOLUTIONS_TO_FIND_ALL_TRIANGULAR_NUMBERS_MULTIPLE_OF_OTHER_TRIANGULAR_NUMBERS/">Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers</a>, 2022.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10,-1,1).

%F Let b(n) be A077290. Then a(n) = (-1 + sqrt(8*b(n) + 1))/2.

%F G.f.: x*(x^3+3*x^2-5*x-3) / ((x-1)*(x^4-10*x^2+1)). - _Colin Barker_, Mar 09 2012

%F From _Vladimir Pletser_, Jul 26 2020: (Start)

%F a(n) = 10*a(n-2) - a(n-4) + 4 with a(-2)=-4, a(-1)=-1, a(0)=0, a(1)=3.

%F Can be defined for negative n by setting a(-n) = - a(n-1) - 1 for all n in Z.

%F a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5). (End)

%e b(3)=630 so a(3) = (-1 + sqrt(8*630 + 1))/2 = (-1 + sqrt(5041))/2 = (71 - 1)/2 = 35.

%p f := gfun:-rectoproc({a(-2) = -4, a(-1) = -1, a(0) = 0, a(1) = 3, a(n) = 10*a(n - 2) - a(n - 4) + 4}, a(n), remember); map(f, [$ (0 .. 40)])[]; # _Vladimir Pletser_, Jul 26 2020

%t LinearRecurrence[{1,10,-10,-1,1},{0,3,8,35,84},30] (* _Harvey P. Dale_, Oct 11 2019 *)

%o (PARI) concat(0, Vec(x*(x^3+3*x^2-5*x-3)/((x-1)*(x^4-10*x^2+1)) + O(x^100))) \\ _Colin Barker_, May 15 2015

%Y Cf. A077288, A077289, A077290.

%K easy,nonn

%O 0,2

%A Bruce Corrigan (scentman(AT)myfamily.com), Nov 03 2002