A156572 Squares of the form k^2+(k+23)^2 with integer k.

289, 529, 1369, 4225, 13225, 42025, 139129, 444889, 1423249, 4721929, 15108769, 48344209, 160402225, 513249025, 1642275625, 5448949489, 17435353849, 55789022809, 185103876169, 592288777609, 1895184495649, 6288082836025

Offset

1,1

Comments

Square roots of k^2+(k+17)^2 are in A156567, values k are in A118337.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,34,-34,0,-1,1).

Formula

a(n) = 34*a(n-3) - a(n-6) - 4232 for n > 6; a(1)=289, a(2)=529, a(3)=1369, a(4)=4225, a(5)=13225, a(6)=42025.

a(n) = A156567(n)^2.

G.f.: x*(289 +240*x +840*x^2 -6970*x^3 +840*x^4 +240*x^5 +289*x^6)/((1-x)*(1 -34*x^3 +x^6)).

Limit_{n -> infinity} a(n)/a(n-3) = 17 + 12*sqrt(2).

Limit_{n -> infinity} a(n)/a(n-1) = ((627 + 238*sqrt(2))/23^2)^2 for n mod 3 = 1.

Limit_{n -> infinity} a(n)/a(n-1) = ((27 + 10*sqrt(2))/23)^2 for n mod 3 = {0, 2}.

a(n) = -289*[n=0] + (529/4) + (3/4)*( f(n/3, 209, 5457)*(n mod 3 = 1) + f((n-1)/3, 209, 1649)*(n mod 3 = 1) + f((n-2)/2, 529, 529)*(n mod 3 = 2) ), where f(n, p, q) = p*ChebyshevU(n, 17) - q*ChebyshevU(n-1, 17). - G. C. Greubel, Jan 04 2022

Example

4225 = 65^2 is of the form k^2+(k+23)^2 with k = 33: 33^2+56^2 = 4225. Hence 4225 is in the sequence.

Mathematica

LinearRecurrence[{1, 0, 34, -34, 0, -1, 1}, {289, 529, 1369, 4225, 13225, 42025, 139129}, 30] (* Harvey P. Dale, Mar 21 2020 *)

Prog

(PARI) {forstep(n=-8, 1800000, [1, 3], if(issquare(a=2*n*(n+23)+529), print1(a, ", ")))}
(Sage)
def f(n, p, q): return p*chebyshev_U(n, 17) - q*chebyshev_U(n-1, 17)
def a(n):
    if (n%3==0): return -289*bool(n==0) + (1/4)*(529 + 3*f(n/3, 209, 5457))
    elif (n%3==1): return (1/4)*(529 + 3*f((n-1)/3, 209, 1649))
    else: return (1/4)*(529 + 3*f((n-2)/3, 529, 529))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

Crossrefs

Cf. A156567, A156575 (first trisection), A156573 (second trisection), A156574 (third trisection).

Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2).

Sequence in context: A008367 A287934 A152852 * A157990 A261111 A218766

Adjacent sequences: A156569 A156570 A156571 * A156573 A156574 A156575

Keyword

nonn,easy

Author

Klaus Brockhaus, Feb 11 2009

Extensions

Revised by Klaus Brockhaus, Feb 16 2009

G.f. corrected, third comment and cross-references edited by Klaus Brockhaus, Sep 22 2009

Status

approved