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 A077399 Triangular numbers that are 1/7 of triangular numbers. 11
 0, 3, 15, 780, 3828, 198135, 972315, 50325528, 246964200, 12782485995, 62727934503, 3246701117220, 15932648399580, 824649301287903, 4046829965558835, 209457675826010160, 1027878878603544528, 53201425010505292755, 261077188335334751295, 13512952494992518349628 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Colin Barker, Table of n, a(n) for n = 0..832 Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021. Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021. Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1). FORMULA Let b(n) be A077398(n) then a(n) = b(n)*(b(n)+1)/2. G.f.: -3*x*(x^2+4*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)). - Colin Barker, Jul 02 2013 Comment from _Vladimir Pletser, Feb 21 2021: (Start) a(n) = 254 a(n - 2) - a (n - 4) + 18. a(n) = a(n - 1) + 254 (a(n - 2) - a(n - 3)) - (a (n - 4) - a(n - 5)). (End) MAPLE f := gfun:-rectoproc({a(-2) = 3, a(-1) = 0, a(0) = 0, a(1) = 3, a(n) = 254*a(n-2)-a(n-4)+18}, a(n), remember); map(f, [`\$`(0 .. 1000)])[] #Vladimir Pletser, Feb 21 2021 MATHEMATICA Select[Accumulate[Range[0, 5700000]], IntegerQ[(Sqrt[56#+1]-1)/2]&] (* Harvey P. Dale, Jan 18 2013 *) LinearRecurrence[{1, 254, -254, -1, 1}, {0, 3, 15, 780, 3828}, 30] (* G. C. Greubel, Jan 18 2018 *) PROG (PARI) concat(0, Vec(-3*x*(x^2+4*x+1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015 (MAGMA) Q:=Rationals(); R:=PowerSeriesRing(Q, 30); [0] cat Coefficients(R!(-3*x*(x^2+4*x+1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)))); // G. C. Greubel, Jan 18 2018 CROSSREFS Cf. A077398, A077400, A077401. Sequence in context: A217449 A167220 A296939 * A219122 A293541 A207332 Adjacent sequences:  A077396 A077397 A077398 * A077400 A077401 A077402 KEYWORD easy,nonn AUTHOR Bruce Corrigan (scentman(AT)myfamily.com), Nov 05 2002 STATUS approved

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Last modified November 28 20:58 EST 2021. Contains 349415 sequences. (Running on oeis4.)