A046184 - OEIS

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A046184

Indices of octagonal numbers which are also squares.

1, 9, 121, 1681, 23409, 326041, 4541161, 63250209, 880961761, 12270214441, 170902040409, 2380358351281, 33154114877521, 461777249934009, 6431727384198601, 89582406128846401, 1247721958419651009, 17378525011746267721, 242051628206028097081 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`The equation a(t)(3a(t)-2) = mm is equivalent to the Pell equation (3a(t)-1)(3a(t)-1) - 3mm = 1. - Paul Weisenhorn, May 12 2009` `As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> infinity} a(n)/a(n-1) = (2 + sqrt(3))^2 = 7 + 4 * sqrt(3). - Ant King, Nov 16 2011` `Also numbers n such that the octagonal number N(n) is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 11 2014` `Also nonnegative integers y in the solutions to 2x^2 - 6y^2 + 4x + 4y + 2 + 2 = 0, the corresponding values of x being A251963. - Colin Barker, Dec 11 2014`
	LINKS	`Vincenzo Librandi, Table of n, a(n) for n = 1..200` `Eric Weisstein's World of Mathematics, Octagonal Square Number.` `Index entries for linear recurrences with constant coefficients, signature (15,-15,1).`
	FORMULA	`{n: A000567(n) in A000290}.` `Nearest integer to (1/6) * (2+sqrt(3))^(2n-1). - Ralf Stephan, Feb 24 2004` `a(n) = A045899(n-1) + 1 = A051047(n+1) + 1 = A003697(2n-2). - N. J. A. Sloane, Jun 12 2004` `a(n) = A001835(n)^2. - Lekraj Beedassy, Jul 21 2006` `From Paul Weisenhorn, May 12 2009: (Start)` `With A=(2+sqrt(3))^2=7+4sqrt(3) the equation xx-3mm=1 has solutions` `x(t) + sqrt(3)m(t) = (2+sqrt(3))A^t and the recurrences` `x(t+2) = 14x(t+1) - x(t) with <x(t)> = 2, 26, 362, 5042` `m(t+2) = 14m(t+1) - m(t) with <m(t)> = 1, 15, 209, 2911` `a(t+2) = 14a(t+1) - a(t) - 4 with <a(t)> = 1, 9, 121, as above. (End)` `From Ant King, Nov 15 2011: (Start)` `a(n) = 14a(n-1) - a(n-2) - 4.` `a(n) = 15a(n-1) - 15a(n-2) + a(n-3).` `a(n) = (1/6)( (2+sqrt(3))^(2n-1) + (2-sqrt(3))^(2n-1) + 2 ).` `a(n) = ceiling( (1/6)(2 + sqrt(3))^(2n-1) ).` `a(n) = (1/6)( (tan(5Pi/12))^(2n-1) + (tan(Pi/12))^(2n-1) + 2 ).` `a(n) = ceiling ( (1/6)(tan(5Pi/12))^(2n-1) ).` `G.f.: x(1-6x+x^2) / ((1-x)(1-14x+x^2)). (End)` `a(n) = A006253(2n-2). - Andrey Goder, Oct 17 2021`
	MATHEMATICA	`LinearRecurrence[ {15, -15, 1}, {1, 9, 121}, 17 ] (* Ant King, Nov 16 2011 )` `CoefficientList[Series[x (1-6x+x^2)/((1-x)(1-14x+x^2)), {x, 0, 30}], x] ( Harvey P. Dale, Sep 01 2021 *)`
	PROG	`(Magma) I:=[1, 9, 121]; [n le 3 select I[n] else 15Self(n-1)-15Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Nov 17 2011` `(PARI) Vec(x(1-6x+x^2) / ((1-x)(1-14x+x^2)) + O(x^100)) \\ Colin Barker, Dec 11 2014`
	CROSSREFS	`Cf. A028230, A036428, A251963.` `Cf. A006253.` `Sequence in context: A302941 A183514 A138978 * A084769 A246467 A202835` `Adjacent sequences: A046181 A046182 A046183 * A046185 A046186 A046187`
	KEYWORD	`nonn,easy`
	AUTHOR	`Eric W. Weisstein`
	STATUS	`approved`

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Last modified December 6 10:23 EST 2022. Contains 358630 sequences. (Running on oeis4.)