This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A077398 #61 Jan 11 2025 05:20:30
%S A077398 0,2,5,39,87,629,1394,10032,22224,159890,354197,2548215,5644935,
%T A077398 40611557,89964770,647236704,1433791392,10315175714,22850697509,
%U A077398 164395574727,364177368759,2620014019925,5803987202642,41755828744080,92499617873520,665473245885362
%N A077398 First member of the Diophantine pair (m,k) that satisfies 7*(m^2+m) = k^2+k; a(n)=m.
%C A077398 Equivalently, m such that 28*m*(m+1)+1 is a square. - _Bruno Berselli_, May 19 2014
%H A077398 Colin Barker, <a href="/A077398/b077398.txt">Table of n, a(n) for n = 0..1000</a>
%H A077398 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 A077398 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 A077398 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 A077398 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,16,-16,-1,1).
%F A077398 G.f.: x*(2+3*x+2*x^2)/((1-x)*(1-16*x^2+x^4)).
%F A077398 a(n) = 16*a(n-2) - a(n-4) + 7, n >= 3. [corrected by _Vladimir Pletser_, Feb 29 2020]
%F A077398 Let b(n) be A077397 then a(n+2) = 2*a(n+1) - a(n) + b(n) with a(0)=0 a(1)=2.
%F A077398 a(0)=0, a(1)=2; a(n+2) = (7 + 16*a(n) + 3*sqrt(1+28*a(n)+28*a(n)^2))/2. - _Herbert Kociemba_, May 12 2008
%F A077398 a(n) = a(n-1) + 16*a(n-2) - 16*a(n-3) - a(n-4) + a(n-5). - _Wesley Ivan Hurt_, Jul 24 2020
%p A077398 f := gfun:-rectoproc({a(-2) = 2, a(-1) = 0, a(0) = 0, a(1) = 2, a(n) = 16*a(n - 2) - a(n - 4) + 7}, a(n), remember): map(f, [$ (0 .. 40)])[]; # _Vladimir Pletser_, Jul 24 2020
%t A077398 LinearRecurrence[{1,16,-16,-1,1},{0,2,5,39,87}, 30] (* _G. C. Greubel_, Jan 18 2018 *)
%t A077398 CoefficientList[Series[x (2+3x+2x^2)/((1-x)(1-16x^2+x^4)),{x,0,40}],x] (* _Harvey P. Dale_, Aug 19 2022 *)
%o A077398 (PARI) {a(n)=if(n<0,0,polcoeff(x*(2+3*x+2*x^2)/((1-x)*(1-16*x^2+x^4)) + x*O(x^n),n))};
%o A077398 (Magma) I:=[0,2,5,39,87]; [n le 5 select I[n] else Self(n-1)+16*Self(n-2) -16*Self(n-3)-Self(n-4)+Self(n-5): n in [1..30]]; // _G. C. Greubel_, Jan 18 2018
%Y A077398 Cf. A077397, A077399, A077400. The k values are in A077401.
%Y A077398 Cf. A053141.
%K A077398 nonn,easy
%O A077398 0,2
%A A077398 Bruce Corrigan (scentman(AT)myfamily.com), Nov 05 2002