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Indices of triangular numbers that are one-tenth of other triangular numbers.
4

%I #43 Mar 31 2023 10:19:50

%S 0,1,6,12,55,246,474,2107,9360,18018,80029,355452,684228,3039013,

%T 13497834,25982664,115402483,512562258,986657022,4382255359,

%U 19463867988,37466984190,166410301177,739114421304,1422758742216,6319209189385,28066884141582,54027365220036,239963538895471,1065802482958830,2051617119619170

%N Indices of triangular numbers that are one-tenth of other triangular numbers.

%C The indices of triangular numbers that are one-tenth of other triangular numbers [t of T(t) such that T(t)=T(u)/10].

%C First member of the Diophantine pair (t, u) that satisfies 10*(t^2 + t) = u^2 + u; a(n) = t.

%C The T(t)'s are in A068085 and the u's are in A341895.

%C Also, nonnegative t such that 40*t^2 + 40*t + 1 is a square.

%C Can be defined for negative n by setting a(n) = a(-1-n) for all n in Z.

%H Vladimir Pletser, <a href="/A341893/b341893.txt">Table of n, a(n) for n = 1..1000</a>

%H Vladimir Pletser, <a href="https://www.researchgate.net/profile/Vladimir-Pletser/publication/359808848_USING_PELL_EQUATION_SOLUTIONS_TO_FIND_ALL_TRIANGULAR_NUMBERS_MULTIPLE_OF_OTHER_TRIANGULAR_NUMBERS/">Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers</a>, 2022.

%H V. Pletser, <a href="https://doi.org/10.1007/s13226-021-00172-y">Recurrent relations for triangular multiples of other triangular numbers</a>, Indian J. Pure Appl. Math. 53 (2022) 782-791

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,38,-38,-1,1).

%F a(n) = (-1 + sqrt(8*b(n) + 1))/2 where b(n) = A068085(n).

%F a(n) = 38 a(n-3) - a(n-6) + 18 for n > 3, with a(-2) = 6, a(-1) = 1, a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 6.

%F a(n) = a(n-1) + 38*(a(n-3) - a(n-4)) - (a(n-6) - a(n-7)) for n >= 4 with a(-2) = 6, a(-1) = 1, a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 6.

%F G.f.: x^2*(1 + 4*x+x^2)*(1+x+x^2)/ ((1-x)*(1-38*x^3+x^6)). - _Stefano Spezia_, Feb 24 2021

%F a(n) = A180003(n) - 1. - _Hugo Pfoertner_, Feb 28 2021

%e a(4) = 12 is a term because its triangular number, (12*13) / 2 = 78 is one-tenth of 780, the triangular number of 39.

%e a(4) = 38 a(1) - a(-2) +18 = 0 - 6 +18 = 12 ;

%e a(5) = 38 a(2) - a(-1) + 18 = 38*1 - 1 +18 = 55.

%p f := gfun:-rectoproc({a(-3) = 6, a(-2) = 1, a(-1) = 0, a(0) = 0, a(1) = 1, a(2) = 6, a(n) = 38*a(n-3)-a(n-6)+18}, a(n), remember); map(f, [`$`(0 .. 1000)])[] ;

%t Rest@ CoefficientList[Series[(x^2*(1 + 5*x + 6*x^2 + 5*x^3 + x^4))/(1 - x - 38*x^3 + 38*x^4 + x^6 - x^7), {x, 0, 31}], x] (* _Michael De Vlieger_, May 19 2022 *)

%Y Cf. A068085, A341895.

%Y 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.

%Y Cf. A180003.

%K easy,nonn

%O 1,3

%A _Vladimir Pletser_, Feb 23 2021