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A156572 Squares of the form k^2+(k+23)^2 with integer k. 3
289, 529, 1369, 4225, 13225, 42025, 139129, 444889, 1423249, 4721929, 15108769, 48344209, 160402225, 513249025, 1642275625, 5448949489, 17435353849, 55789022809, 185103876169, 592288777609, 1895184495649, 6288082836025 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

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

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

LINKS

Table of n, a(n) for n=1..22.

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.

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)).

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, ", ")))}

CROSSREFS

Equals A156567^2. Cf. 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

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Last modified April 14 10:14 EDT 2021. Contains 342949 sequences. (Running on oeis4.)