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 A319170 Triangular numbers of the form 2..21..1; n_times 2 followed with n_times 1; n >= 1. 3
 21, 2211, 222111, 22221111, 2222211111, 222222111111, 22222221111111, 2222222211111111, 222222222111111111, 22222222221111111111, 2222222222211111111111, 222222222222111111111111, 22222222222221111111111111, 2222222222222211111111111111, 222222222222222111111111111111, 22222222222222221111111111111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Triangular numbers of the  form (5^(2x)*2^(2x+1)-10^x-1)/9. - Harvey P. Dale, Sep 16 2019 LINKS Colin Barker, Table of n, a(n) for n = 1..500 Jiri Sedlacek, Trojuhelnikova cisla, In: Jiří Sedláček (author): Faktoriály a kombinační čísla. (Czech). Praha: Mladá fronta, 1964. pp. 60-71. Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000). FORMULA For n >= 1, a(n) = 2..21..1; n_times 2 followed with n_times 1. a(n) = A000217(n_times 6), that is a(n) = A000217(A002280(n)). a(n) = 1/9 * (2*10^n + 1) * (10^n - 1), that is a(n) = 1/9 * A199682(n) * A002283(n). From Colin Barker, Sep 13 2018: (Start) G.f.: 3*x*(7 - 40*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)). a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3 (End) EXAMPLE a(1) = A000217(6) = 21; a(2) = A000217(66) = 2211; a(3) = A000217(666) = 222111. MATHEMATICA Select[Table[FromDigits[Join[PadRight[{}, n, 2], PadRight[{}, n, 1]]], {n, 20}], OddQ[ Sqrt[8#+1]]&] (& or *) Select[Table[(5^(2x) 2^(2x+1)-10^x-1)/9, {x, 20}], OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Sep 16 2019 *) PROG (PARI) Vec(3*x*(7 - 40*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20)) \\ Colin Barker, Sep 13 2018 CROSSREFS Cf. A000217, A002280, A002283, A199682. Sequence in context: A219104 A219983 A119099 * A215160 A250062 A219000 Adjacent sequences:  A319167 A319168 A319169 * A319171 A319172 A319173 KEYWORD nonn,base AUTHOR Ctibor O. Zizka, Sep 12 2018 STATUS approved

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Last modified August 6 22:02 EDT 2020. Contains 336256 sequences. (Running on oeis4.)
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