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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

COMMENTS

a(n) = (b(n)-1)^2 + b(n)^2 + (b(n)+1)^2 = c(n)^2 + (c(n)+1)^2, where b(n) is A054320 and c(n) is A031138; a(n) = 3b(n)+2, where b(n) is a Star square number (A006061).

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

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Index entries for sequences related to sums of squares

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

FORMULA

A007667 = 3*Square star numbers (A006061) + 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.

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

Cf. A003154, A031138, A006061, A054320.

Sequence in context: A006430 A301613 A180766 * A247082 A121668 A234311

Adjacent sequences:  A007664 A007665 A007666 * A007668 A007669 A007670

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

Additional comments from Ignacio Larrosa CaƱestro Feb 27 2000

Corrected by T. D. Noe, Nov 07 2006

STATUS

approved

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Last modified October 21 04:26 EDT 2019. Contains 328291 sequences. (Running on oeis4.)