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A007667 The sum of both two and three consecutive squares.
(Formerly M4037)
8
5, 365, 35645, 3492725, 342251285, 33537133085, 3286296790925, 322023548377445, 31555021444198565, 3092070077983081805, 302991312620897818205, 29690056566770003102165 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
FORMULA
From Ignacio Larrosa Cañestro, Feb 27 2000: (Start)
a(n) = (b(n)-1)^2 + b(n)^2 + (b(n)+1)^2 = c(n)^2 + (c(n)+1)^2, where b(n) = A054320(n) and c(n) = A031138(n).
a(n) = 3*A006061(n) + 2.
a(n) = 99*(a(n-1) - a(n-2)) + a(n-3).
a(n) = 3*(5 - 2*sqrt(6))/8*(sqrt(3) + sqrt(2))^(4*n) + 3*(5 + 2*sqrt(6))/8*(sqrt(3) - sqrt(2))^(4*n) + 5/4. (End)
G.f.: 5*x*(1-26*x+x^2)/((1-x)*(1-98*x+x^2)). - Colin Barker, Apr 14 2012
EXAMPLE
a(2) = 365 = 13^2+14^2 = 10^2+11^2+12^2.
MATHEMATICA
CoefficientList[Series[5*(1-26*x+x^2)/((1-x)*(1-98*x+x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 16 2012 *)
PROG
(PARI) my(x='x+O('x^20)); Vec(5*x*(1-26*x+x^2)/((1-x)*(1-98*x+x^2))) \\ G. C. Greubel, Jul 23 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 5*x*(1-26*x+x^2)/((1-x)*(1-98*x+x^2)) )); // G. C. Greubel, Jul 23 2019
(Sage) (5*x*(1-26*x+x^2)/((1-x)*(1-98*x+x^2))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019
(GAP) a:=[5, 365, 35645];; for n in [4..20] do a[n]:=99*a[n-1]-99*a[n-2] + a[n-3]; od; a; # G. C. Greubel, Jul 23 2019
CROSSREFS
Sequence in context: A006430 A301613 A180766 * A247082 A121668 A234311
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected by T. D. Noe, Nov 07 2006
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)