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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
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
Sequence in context: A370678 A211914 A194792 * A099128 A172958 A286397
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 March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)