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A133217
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Indices of decagonal numbers (A001107) that are also triangular (A000217).
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2
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0, 1, 2, 20, 55, 667, 1856, 22646, 63037, 769285, 2141390, 26133032, 72744211, 887753791, 2471161772, 30157495850, 83946756025, 1024467105097, 2851718543066, 34801724077436, 96874483708207, 1182234151527715, 3290880727535960, 40161159427864862
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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FORMULA
| For n>5, a(n) = 34*a(n-2) - a(n-4) - 12.
For n>6, a(n) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5).
For n>1, a(n) = 1/16 * ((2*sqrt(2) + (-1)^n)*(1 + sqrt(2))^(2*n - 3) - (2*sqrt(2) - (-1)^n)*(1 - sqrt(2))^(2*n - 3) + 6).
For n>1, a(n) = ceiling (1/16*(2*sqrt(2) + (-1)^n)*(1 + sqrt(2))^(2*n - 3)).
GF: ( 1 - 33*x^2 + 18*x^3 + 2*x^4 ) / ((1 - x ) * (1 - 6*x + x^2 ) * (1 + 6*x + x^2)).
lim (n -> Infinity, a(2n+1)/a(2n)) = 1/7*(43 + 30*sqrt(2)).
lim (n -> Infinity, a(2n)/a(2n-1)) = 1/7*(11 + 6*sqrt(2)).
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EXAMPLE
| The third number which is both decagonal (A001107) and triangular (A000217) is A133216(3)=10. As this is the second decagonal number, we have a(3) = 2.
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MATHEMATICA
| LinearRecurrence[{1, 34, -34, -1, 1} , {1, 2, 20, 55, 667}, 23]
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CROSSREFS
| Cf. A000217, A001107, A077443, A077442, A133216, A133218.
Sequence in context: A097652 A059211 A139271 * A001504 A192351 A136905
Adjacent sequences: A133214 A133215 A133216 * A133218 A133219 A133220
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KEYWORD
| nonn
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AUTHOR
| Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 2007; Ant King (mathstutoring(AT)ntlworld.com), Nov 04 2011
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EXTENSIONS
| Entry revised by Max Alekseyev (maxale(AT)gmail.com), Nov 06 2011
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