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 A217758 Triangular numbers of the form k^2 + k - 1. 0
 1, 55, 1891, 64261, 2183005, 74157931, 2519186671, 85578188905, 2907139236121, 98757155839231, 3354836159297755, 113965672260284461, 3871478020690373941, 131516287031212429555, 4467682281040532230951, 151769681268346883422801 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Triangular numbers belonging to A028387. - Bruno Berselli, Apr 18 2018 Since this sequence lists triangular numbers t such that 4*t + 5 = u^2 and 8*t + 1 = v^2 by definition of triangular numbers, 2*u^2 - 9 = 2*(4*t + 5) - 9 = 8*t + 1 = v^2, that is, sqrt(4*a(n) + 5) is in A075841. - Altug Alkan, Apr 18 2018 LINKS Table of n, a(n) for n=1..16. Index entries for linear recurrences with constant coefficients, signature (35,-35,1). FORMULA From Giovanni Resta, Mar 24 2013: (Start) G.f.: -x*(1 + 20*x + x^2) / ( (x - 1)*(x^2 - 34*x + 1) ). a(n) = floor(9 * (17 + sqrt(288))^n * (3 - sqrt(8))/32). (End) a(n) = (A075841(n)^2 - 5)/4. - Altug Alkan, Apr 18 2018 EXAMPLE Triangular(10) = 55 = 7^2 + 7 - 1, so 55 is in the sequence. MATHEMATICA a[1]=1; a[2]=55; a[3]=1891; a[n_] := a[n] = 35*a[n-1] - 35*a[n-2] + a[n-3]; Array[a, 20] (* Giovanni Resta, Mar 24 2013 *) Table[Floor@(9 (17 + Sqrt@ 288)^n*(3 - Sqrt@ 8)/32), {n, 0, 16}] (* or *) CoefficientList[Series[-x (1 + 20 x + x^2)/((x - 1) (x^2 - 34 x + 1)), {x, 0, 16}], x] (* Michael De Vlieger, Oct 08 2016 *) PROG (C) #include typedef unsigned long long U64; U64 rootPronic(U64 a) { U64 sr = 1L<<32, s, b; while (a < sr*(sr-1)) sr>>=1; for (b = sr>>1; b; b>>=1) { s = sr+b; if (a >= s*(s-1)) sr = s; } return sr; } int main() { U64 a, i, t; for (i=0; i < 1L<<32; ++i) { a = i*(i+1)/2 + 1; t = rootPronic(a); if (a == t*(t-1)) printf("%llu %llu %llu\n", i, t, a-1); } return 0; } (PARI) Vec(x*(1+20*x+x^2)/(1-35*x+35*x^2-x^3)+O(x^66)) \\ Joerg Arndt, Mar 25 2013 CROSSREFS Cf. A000217, A028387, A069017, A075841. Sequence in context: A163722 A321034 A275942 * A346325 A240687 A053113 Adjacent sequences: A217755 A217756 A217757 * A217759 A217760 A217761 KEYWORD nonn,easy AUTHOR Alex Ratushnyak, Mar 23 2013 STATUS approved

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Last modified June 25 10:28 EDT 2024. Contains 373701 sequences. (Running on oeis4.)