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A133217 Indices of decagonal numbers (A001107) that are also triangular (A000217). 2
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; text; internal format)
OFFSET

1,3

COMMENTS

For n>0, a(n) = (A055979(n) - A056161(n))/2, with those two sequences related through the Diophantine equation 2x^2 + 3x + 2 = r^2. - Richard R. Forberg, Nov 24 2013

LINKS

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (1, 34, -34, -1, 1).

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)).

G.f.: ( 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)).

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.

MATHEMATICA

LinearRecurrence[{1, 34, -34, -1, 1} , {0, 1, 2, 20, 55, 667}, 24] (* first term 0 corrected by Georg Fischer, Apr 02 2019 *)

CROSSREFS

Cf. A000217, A001107, A077443, A077442, A133216, A133218.

Sequence in context: A225065 A059211 A139271 * A001504 A192351 A136905

Adjacent sequences:  A133214 A133215 A133216 * A133218 A133219 A133220

KEYWORD

nonn

AUTHOR

Richard Choulet, Oct 11 2007; Ant King, Nov 04 2011

EXTENSIONS

Entry revised by Max Alekseyev, Nov 06 2011

STATUS

approved

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Last modified November 14 04:53 EST 2019. Contains 329108 sequences. (Running on oeis4.)