The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A341893 Indices of triangular numbers that are one-tenth of other triangular numbers. 4
 0, 1, 6, 12, 55, 246, 474, 2107, 9360, 18018, 80029, 355452, 684228, 3039013, 13497834, 25982664, 115402483, 512562258, 986657022, 4382255359, 19463867988, 37466984190, 166410301177, 739114421304, 1422758742216, 6319209189385, 28066884141582, 54027365220036, 239963538895471, 1065802482958830, 2051617119619170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The indices of triangular numbers that are one-tenth of other triangular numbers [t of T(t) such that T(t)=T(u)/10]. First member of the Diophantine pair (t, u) that satisfies 10*(t^2 + t) = u^2 + u; a(n) = t. The T(t)'s are in A068085 and the u's are in A341895. Also, nonnegative t such that 40*t^2 + 40*t + 1 is a square. Can be defined for negative n by setting a(n) = a(-1-n) 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) = (-1 + sqrt(8*b(n) + 1))/2 where b(n) = A068085(n). 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. 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. G.f.: (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). - Stefano Spezia, Feb 24 2021 a(n) = A180003(n) - 1. - Hugo Pfoertner, Feb 28 2021 EXAMPLE a(4) = 12 is a term because its triangular number, (12*13) / 2 = 78 is one-tenth of 780, the triangular number of 39. a(4) = 38 a(1) - a(-2) +18 = 0 - 6 +18 = 12 ; a(5) = 38 a(2) - a(-1) + 18 = 38*1 - 1 +18 = 55. MAPLE 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)])[] ; MATHEMATICA 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 *) CROSSREFS Cf. A068085, A341895. 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. Cf. A180003. Sequence in context: A117866 A290999 A094060 * A324529 A076722 A322288 Adjacent sequences: A341890 A341891 A341892 * A341894 A341895 A341896 KEYWORD easy,nonn AUTHOR Vladimir Pletser, Feb 23 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 11:09 EST 2022. Contains 358493 sequences. (Running on oeis4.)