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

Offset: 1

Author

Colin Barker, Jan 02 2015

Keywords

• nonn
• easy

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

Examples

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

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).

Crossrefs

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

Programs

Mathematica

LinearRecurrence[{11, -11, 1}, {1, 6, 55}, 25] (* Paolo Xausa, May 30 2025 *)

PARI

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

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