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 A341895 Indices of triangular numbers that are ten times other triangular numbers. 5
 0, 4, 20, 39, 175, 779, 1500, 6664, 29600, 56979, 253075, 1124039, 2163720, 9610204, 42683900, 82164399, 364934695, 1620864179, 3120083460, 13857908224, 61550154920, 118481007099, 526235577835, 2337285022799, 4499158186320, 19983094049524, 88755280711460, 170849530073079, 758831338304095, 3370363382012699 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Second member of the Diophantine pair (b(n), a(n)) that satisfies a(n)^2 + a(n) = 10*(b(n)^2 + b(n)) or T(a(n)) = 10*T(b(n)) where T(x) is the triangular number of x. The T(b)'s are in A068085 and the b's are in A341893. Can be defined for negative n by setting a(-n) = -a(n+1) - 1 for all n in Z. LINKS Vladimir Pletser, Table of n, a(n) for n = 1..1000 Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2022. Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1). FORMULA a(n) = 38*a(n-3) - a(n-6) + 18 for n > 3, with a(-2) = -21, a(-1) = -5, a(0) = -1, a(1) = 0, a(2) = 4, a(3) = 20. a(n) = a(n-1) + 38*(a(n-3) - a(n-4)) - (a(n-6) - a(n-7)) for n >= 4 with a(-2) = -21, a(-1) = -5, a(0) = -1, a(1) = 0, a(2) = 4, a(3) = 20. G.f.: x^2*(4 + 16*x + 19*x^2 - 16*x^3 - 4*x^4 - x^5)/(1 - x - 38*x^3 + 38*x^4 + x^6 - x^7). - Stefano Spezia, Feb 24 2021 a(n) = (A198943(n) + 1)/2 - 1. - Hugo Pfoertner, Feb 26 2021 EXAMPLE a(2) = 4 is a term because its triangular number, T(a(2)) = 4*5 / 2 = 10 is ten times a triangular number. a(4) = 38*a(1) - a(-2) + 18 = 38*0 - (-21) + 18 = 39, etc. MAPLE f := gfun:-rectoproc({a(-2) = -21, a(-1) = -5, a(0) = -1, a(1) = 0, a(2) = 4, a(3) = 20, a(n) = 38*a(n-3)-a(n-6)+18}, a(n), remember); map(f, [`\$`(0 .. 1000)]) ; # MATHEMATICA Rest@ CoefficientList[Series[x^2*(4 + 16*x + 19*x^2 - 16*x^3 - 4*x^4 - x^5)/(1 - x - 38*x^3 + 38*x^4 + x^6 - x^7), {x, 0, 30}], x] (* Michael De Vlieger, May 19 2022 *) CROSSREFS Cf. A341893, A068085, A166477 (n=10). Cf. A336623, A336624, A336626, A336625, A053141, A001652, A075528, A029549, A061278, A001571, A076139, A076140, A077259, A077262, A077260, A077261, A077288, A077291, A077289, A077290, A077398, A077401, A077399, A077400, A000217. Sequence in context: A232221 A323183 A072977 * A163365 A145194 A164924 Adjacent sequences: A341892 A341893 A341894 * A341896 A341897 A341898 KEYWORD easy,nonn AUTHOR Vladimir Pletser, Feb 23 2021 STATUS approved

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Last modified May 29 17:30 EDT 2023. Contains 363042 sequences. (Running on oeis4.)