0,2

Solution to a(a-1) = 2b(b-1) in natural numbers: a = a(n), b = b(n) = A011900(n).

n such that n^2 = (1/2)*(n+floor(sqrt(2)*n*floor(sqrt(2)*n))). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 15 2003

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.

T. W. Forget and T. A. Larkin, Pythagorean triads of the form X, X+1, Z described by recurrence sequences, Fib. Quart., 6 (No. 3, 1968), 94-104.

L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

Ron Knott, Pythagorean Triples and Online Calculators

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a(n) = (1+sqrt(1+8*b(n)*(b(n)+1)))/2 with b(n) = A011900(n).

a(n) = 6*a(n-1)-a(n-2)-2, n >= 2, a(0) = 1, a(1) = 4. a(n) = (A(n+1)-3*A(n)+2)/4 with A(n) = A001653(n).

G.f.: (1-3*x)/((1-6*x+x^2)*(1-x)). a(n) = partial sums of A001541(n). - Barry Williams, May 03 2000

A001652(n)*A001652(n+1) + a(n)*a(n+1) = A001542(n+1)^2 = A084703(n+1) - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003

a(n) = 1/2 + ((1-2^{1/2})/4)*(3 - 2^{3/2})^n + ((1+2^{1/2})/4)*(3 + 2^{3/2})^n. - Antonio Olivares (olivares14031(AT)yahoo.com), Oct 13, 2003

Let a(n) = A001652(n), b(n) = this sequence and c(n) = A001653(n). Then for k>j, c(i)*(c(k) - c(j)) = a(k+i)+...+a(i+j+1) + a(k-i-1)+...+a(j-i) + k - j. For n<0, a(n) = -b(-n-1). Also a(n)*a(n+2k+1) + b(n)*b(n+2k+1) + c(n)*c(n+2k+1) = (a(n+k+1) - a(n+k))^2; a(n)*a(n+2k) + b(n)*b(n+2k) + c(n)*c(n+2k) = 2*c(n+k)^2 - Charlie Marion (charliem(AT)bestweb.net), Jul 01 2003

2*a(n)=2*A084159(n) + 1 + (-1)^(n+1)=2*A046729(n) + 1 - (-1)^(n+1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 16 2004

(PARI) a(n)=(2-subst(poltchebi(abs(n))-poltchebi(abs(n+1)), x, 3))/4

Other 2 sides are A001652 and A001653.

Cf. A011900, A001541. A001652(n)=-a(-1-n).

Sequence in context: A024051 A020048 A093426 this_sequence A045721 A101810 A001888

Adjacent sequences: A046087 A046088 A046089 this_sequence A046091 A046092 A046093

nonn,easy,nice

Eric Weisstein (eric(AT)weisstein.com)

Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)