A253475 - OEIS

A253475

Indices of centered square numbers (A001844) which are also centered hexagonal numbers (A003215).

1, 6, 55, 540, 5341, 52866, 523315, 5180280, 51279481, 507614526, 5024865775, 49741043220, 492385566421, 4874114620986, 48248760643435, 477613491813360, 4727886157490161, 46801248083088246, 463284594673392295, 4586044698650834700, 45397162391834954701

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OFFSET

1,2

COMMENTS

Also positive integers x in the solutions to 4*x^2 - 6*y^2 - 4*x + 6*y = 0, the corresponding values of y being A054318.

Also indices of centered hexagonal numbers (A003215) which are also hexagonal numbers (A000384).

Also indices of terms in sequence A193218 which are the square root of a sum of 5th powers (A000539). - Daniel Poveda Parrilla, Jun 10 2017

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).

Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).

Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).

Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).

Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

FORMULA

a(n) = 11*a(n-1)-11*a(n-2)+a(n-3).

G.f.: x*(5*x-1) / ((x-1)*(x^2-10*x+1)).

a(n) = sqrt((-2-(5-2*sqrt(6))^n-(5+2*sqrt(6))^n)*(2-(5-2*sqrt(6))^(1+n)-(5+2*sqrt(6))^(1+n)))/(4*sqrt(2)). - Gerry Martens, Jun 04 2015

2*a(n) = 1+A054320(n-1). - R. J. Mathar, Feb 07 2022

EXAMPLE

6 is in the sequence because the 6th centered square number is 61, which is also the 5th centered hexagonal number.

PROG

(PARI) Vec(x*(5*x-1)/((x-1)*(x^2-10*x+1)) + O(x^100))

CROSSREFS

Cf. A000384, A000539, A001844, A003215, A054318, A193218, A253175.