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A131750 Numbers that are both centered triangular and centered square. 4
1, 85, 16381, 3177721, 616461385, 119590330861, 23199907725541, 4500662508423985, 873105326726527441, 169377932722437899461, 32858445842826225967885, 6374369115575565399870121 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

We solve r^2+(r+1)^2=0.5*(3*p^2+3*p+2), which is equivalent to (4*r+2)^2=3*(2*p+1)^2+1.

The Diophantine equation X^2=3*Y^2+1 gives X by A001075 and Y by A013453. The return to r gives the sequence 0,6,90,1260,17556,... which satisfies the formulas a(n+2)=14*a(n+1)-a(n)+6 and a(n+1)=7*a(n)+3+(48*a(n)^2+48*a(n)+9)^0.5 and the return to p the sequence A001921 which satisfies this new relation: a(n+1)=7*a(n)+sqrt(48*a(n)^2+48*a(n)+16). Then we obtain the present sequence.

LINKS

Robert Israel, Table of n, a(n) for n = 1..394

Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

FORMULA

a(n+2) = 195*a(n+1)-195*a(n)+a(n-1).

a(n+1) = 97*a(n) - 54 + 14*sqrt(48*a(n)^2-54*a(n)+15).

G.f.: x*(1-110*x+x^2)/((1-x)*(1-194*x+x^2)).

a(n) = 9/16 + (7/32)*(97-56*sqrt(3))^n + (7/32)*(97+56*sqrt(3))^n - (1/8)*(97-56*sqrt(3))^n*sqrt(3) + (1/8)*sqrt(3)*(97+56*sqrt(3))^n, with n>=0. - Paolo P. Lava, Sep 26 2008

MAPLE

A131750 := proc(n) coeftayl(x*(1-110*x+x^2)/(1-x)/(1-194*x+x^2), x=0, n) ; end: seq(A131750(n), n=1..20) ; # R. J. Mathar, Oct 24 2007

MATHEMATICA

LinearRecurrence[{195, -195, 1}, {1, 85, 16381}, 20] (* Harvey P. Dale, Apr 26 2018 *)

PROG

(MAGMA) [n le 2 select 1 else Floor(97*Self(n-2) - 54 + 14*Sqrt(48*Self(n-2)^2-54*Self(n-2)+15)): n in [2..30]]; // Vincenzo Librandi, Aug 26 2015

CROSSREFS

Cf. A001921, A001075, A001353.

Intersection of A001844 and A005448.

Sequence in context: A006106 A015338 A181015 * A239269 A068749 A172826

Adjacent sequences:  A131747 A131748 A131749 * A131751 A131752 A131753

KEYWORD

nonn

AUTHOR

Richard Choulet, Sep 20 2007

EXTENSIONS

More terms from R. J. Mathar, Oct 24 2007

Recurrences corrected by Robert Israel, Aug 26 2015

Name corrected by Daniel Poveda Parrilla, Sep 19 2016

STATUS

approved

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Last modified April 22 08:56 EDT 2019. Contains 322329 sequences. (Running on oeis4.)