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 A129803 Triangular numbers that are the sum of three consecutive triangular numbers. 19
 10, 136, 1891, 26335, 366796, 5108806, 71156485, 991081981, 13803991246, 192264795460, 2677903145191, 37298379237211, 519499406175760, 7235693307223426, 100780206894952201, 1403687203222107385, 19550840638214551186, 272308081731781609216 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Indices m: 4, 16, 61, 229, 856, 3196, 11929, with recurrence m(i) = 5(m(i-1) - m(i-2)) + m(i-3) (see A133161). If first term is omitted, same sequence as A128862. - R. J. Mathar, Jun 13 2008 LINKS Colin Barker, Table of n, a(n) for n = 1..850 Index entries for linear recurrences with constant coefficients, signature (15, -15, 1). FORMULA a(n) = tr(m) = tr(k) + tr(k+1) + tr(k+2), where tr(k) = k(k+1)/2 = A000217(k). From Richard Choulet, Oct 06 2007: (Start) a(n+2) = 14*a(n+1) - a(n) - 3. a(n+1) = 7*a(n) - 3/2 + 1/2*sqrt(192*a(n)^2 - 96*a(n) - 15). G.f.: x*(10-14*x+x^2) / ((1-x)*(1-14*x+x^2)). (End) a(n) = (4-3*(7-4*sqrt(3))^n*(-2+sqrt(3))+3*(2+sqrt(3))*(7+4*sqrt(3))^n)/16. - Colin Barker, Mar 05 2016 EXAMPLE With tr(k) = k(k+1)/2 = A000217(k): 10 = tr(4) = tr(1) + tr(2) + tr(3) = 1 + 3 + 6, 136 = tr(16) = tr(8) + tr(9) + tr(10) = 36 + 45 + 55, 1891 = tr(61) = tr(34) + tr(35) + tr(36) = 595 + 630 + 666, 26335 = tr(229) = tr(131) + tr(132) + tr(133) = 8646 + 8778 + 8911, 366796 = tr(856) = tr(493) + tr(494) + tr(495) = 121771 + 122265 + 122760. PROG (PARI) Vec((10*z - 14*z^2 + z^3)/((1-z)*(1 - 14*z + z^2)) + O(z^30)) \\ Michel Marcus, Sep 16 2015 CROSSREFS Cf. A000217, A128862, A133161. Sequence in context: A240917 A240654 A128862 * A065024 A026244 A261503 Adjacent sequences:  A129800 A129801 A129802 * A129804 A129805 A129806 KEYWORD nonn,easy AUTHOR Zak Seidov, May 18 2007 STATUS approved

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Last modified August 4 18:15 EDT 2021. Contains 346455 sequences. (Running on oeis4.)