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A076293
Numbers k where the root mean square (RMS) of k and 7 is an integer, i.e., sqrt((k^2 + 7^2)/2) is an integer.
4
1, 7, 17, 23, 49, 103, 137, 287, 601, 799, 1673, 3503, 4657, 9751, 20417, 27143, 56833, 118999, 158201, 331247, 693577, 922063, 1930649, 4042463, 5374177, 11252647, 23561201, 31322999, 65585233, 137324743, 182563817, 382258751, 800387257, 1064059903
OFFSET
0,2
LINKS
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
FORMULA
a(n) = 6a(n-3) - a(n-6) = sqrt(2*A076294(n)^2 - 49) = A076295(n) + A076296(n).
a(3n+1) = 7*A002315(n).
G.f.: (x+1)*(x^2+3*x+1)^2 / (x^6-6*x^3+1). - Colin Barker, Sep 14 2014
EXAMPLE
17 is in the sequence since sqrt((17^2 + 7^2)/2) = 13 is an integer.
MATHEMATICA
Column[LinearRecurrence[{0, 0, 6, 0, 0, -1}, {1, 7, 17, 23, 49, 103}, 35] ] (* Vincenzo Librandi, Jul 30 2017 *)
PROG
(PARI) Vec((x+1)*(x^2+3*x+1)^2/(x^6-6*x^3+1) + O(x^100)) \\ Colin Barker, Sep 14 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Oct 05 2002
EXTENSIONS
More terms from Colin Barker, Sep 14 2014
STATUS
approved