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 A133216 Integers that are simultaneously triangular (A000217) and decagonal (A001107). 2
 0, 1, 10, 1540, 11935, 1777555, 13773376, 2051297326, 15894464365, 2367195337045, 18342198104230, 2731741367653000, 21166880717817451, 3152427171076225351, 24426562006163234620, 3637898223680596402450, 28188231388231654934425, 4198131397700237172202345 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Positive terms are of the form (m^2-9)/16 where m runs over the elements of A077443 that are congruent to 5 modulo 8. Correspondingly, for n>1, sqrt(16*a(n)+9) form a subsequence of A077443, while sqrt(8*a(n)+1) form a subsequence of A077442 with indices congruent to 2,3 modulo 4. [Max Alekseyev] LINKS Index entries for linear recurrences with constant coefficients, signature (1, 1154, -1154, -1, 1). FORMULA a(n) = A000217(A133218(n)) = A001107(A133217(n)). For n>5, a(n) = 1154*a(n-2) - a(n-4) + 396. For n>6, a(n) = a(n-1) + 1154*a(n-2) - 1154*a(n-3) - a(n-4) + a(n-5). For n>1, a(n) = 1/64 * ( (9 + 4* sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-6) + (9 - 4* sqrt(2)*(-1)^n)*(1-sqrt(2))^(4*n-6) - 22). a(n) = floor ( 1/64 * (9 + 4*sqrt(2)*(-1)^n) * (1+sqrt(2))^(4*n-6) ). G.f.: (x^5 + 9*x^4 + 376*x^3 + 9*x^2 + x)/((1 - x)*(x^2 - 34*x + 1)*(x^2 + 34*x + 1)). [corrected by Peter Luschny, Apr 04 2019] Lim (n -> Infinity, a(2n+1)/a(2n)) = (1/49)*(3649+2580*sqrt(2)). Lim (n -> Infinity, a(2n)/a(2n-1)) = (1/49)*(193+132*sqrt(2)). EXAMPLE The initial terms of the sequences of triangular (A000217) and decagonal (A001107) numbers are 0, 1, 3, 6, 10, 15, ... and 0, 1, 10, 27, ... respectively. As the third number which is common to both sequences is 10, we have a(3) = 10. MATHEMATICA LinearRecurrence[{1, 1154, -1154, -1, 1} , {0, 1, 10, 1540, 11935, 1777555}, 17] (* first term 0 corrected by Georg Fischer, Apr 02 2019 *) CROSSREFS Cf. A000217, A001107, A133217, A133218, A077443, A077442. Sequence in context: A160104 A211914 A194792 * A099128 A172958 A286397 Adjacent sequences:  A133213 A133214 A133215 * A133217 A133218 A133219 KEYWORD nonn AUTHOR Richard Choulet, Oct 11 2007; Ant King, Nov 04 2011 EXTENSIONS Entry revised by N. J. A. Sloane, Nov 06 2011 Term 0 prepended and entry revised accordingly by Max Alekseyev, Nov 06 2011 STATUS approved

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Last modified December 6 16:24 EST 2019. Contains 329808 sequences. (Running on oeis4.)