A111766 Numbers occurring in three Pythagorean triples of the form: odd: a, (a^2-1)/2, (a^2+1)/2 or even: a, a^2/4-1, a^2/4+1.

0, 5, 24, 145, 840, 4901, 28560, 166465, 970224, 5654885, 32959080, 192099601, 1119638520, 6525731525, 38034750624, 221682772225, 1292061882720, 7530688524101, 43892069261880, 255821727047185, 1491038293021224, 8690408031080165, 50651409893459760

Offset

1,2

Comments

This parallels Cassini's identity for Fibonacci numbers (Mathworld).

Links

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

Eric Weisstein's World of Mathematics, Cassini's Identity.

Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Formula

a(n) = A000129(n-1)*A000129(n+1) = A000129(n)^2 + (-1)^n.

G.f.: -x^2*(-5+x) / ( (1+x)*(1-6*x+x^2) ). - R. J. Mathar, Sep 21 2011

a(n) = A001333(n)^2 - A000129(n)^2 for n >= 1. - Richard R. Forberg, Aug 24 2013

From Colin Barker, Nov 04 2016: (Start)

a(n) = (6*(-1)^n+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/8 for n > 0.

a(n) = 5*a(n-1)+5*a(n-2)-a(n-3) for n > 3. (End)

Sum_{n>=2} 1/a(n) = 1/4. - Amiram Eldar, Dec 28 2025

Example

a(5) = P(4)*P(6) = 12*70 = 840 = P(5)-1 = 29^2-1.

Mathematica

LinearRecurrence[{5, 5, -1}, {0, 5, 24}, 25] (* Amiram Eldar, Dec 28 2025 *)

Prog

(PARI) concat(0, Vec(-x^2*(-5+x)/((1+x)*(1-6*x+x^2)) + O(x^30))) \\ Colin Barker, Nov 04 2016

Crossrefs

Cf. A000129, A001333, A076218, A078522 (bisections).

Sequence in context: A232318 A201952 A221788 * A368534 A228067 A372138

Adjacent sequences: A111763 A111764 A111765 * A111767 A111768 A111769

Keyword

nonn,easy

Author

Jeremy C. Buchanan (jbuchanan(AT)myhww.org), Nov 21 2005

Status

approved