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A055997 Numbers n such that n(n-1)/2 is a square. 15
1, 2, 9, 50, 289, 1682, 9801, 57122, 332929, 1940450, 11309769, 65918162, 384199201, 2239277042, 13051463049, 76069501250, 443365544449, 2584123765442, 15061377048201, 87784138523762, 511643454094369, 2982076586042450, 17380816062160329, 101302819786919522 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Numbers n such that (n-th triangular number - n) is a square.

Gives solutions to A007913(2x)=A007913(x-1). - Benoit Cloitre, Apr 07 2002

Number of closed walks of length 2n on the grid graph P_2 X P_3. - Mitch Harris, Mar 06 2004

a(2k) = A001541(k)^2. - Alexander Adamchuk, Nov 24 2006

If x=A001109(n-1), y=a(n) and z=x^2+y, then x^4+y^3=z^2. - Bruno Berselli, Aug 24 2010

The product of any term a(n) with an even successor a(n+2k) is always a square number. The product of any term a(n) with an odd successor a(n+2k+1) is always twice a square number. - Bradley Klee & Bill Gosper, Jul 22 2015

It appears that dividing even terms by two and taking sqrt gives sequence A079496. - Bradley Klee, Jul 25 2015

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

LINKS

Colin Barker, Table of n, a(n) for n = 1..100

Dario Alpern, a^4+b^3=c^2.

P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 93-105.

K. Ramsey, Generalized Proof re Square Triangular Numbers

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

FORMULA

a(n) = 6*a(n-1)-a(n-2)-2; n >= 3, a(1) = 1, a(2) = 2.

G.f.: x*(1-5*x+2*x^2)/((1-x)*(1-6*x+x^2)).

a(n)-1+sqrt(2*a(n)*(a(n)-1)) = A001652(n); e.g., 50-1+(2*50*49)^0.5 = 119. - Charlie Marion, Jul 21 2003

a(n) = IF(mod(n; 2)=0; (((1-sqrt(2))^n+(1+sqrt(2))^n)/2)^2; 2*((((1-sqrt(2))^(n+1)+(1+sqrt(2))^(n+1))-(((1-sqrt(2))^n+(1+sqrt(2))^n)))/4)^2). The even-indexed terms are a(n) = [A001333(n)]^2; the odd-indexed terms are a(n) = 2*[ [A001333(n+1) - A001333(n)]/4 ]^2 = 2*[ [A001333(n+1) - A001333(n)]/4 ]^2 = 2*[A001653(n)]^2. - Antonio Alberto Olivares, Jan 31 2004

A053141(n+1) + a(n+1) = A001541(n+1) + A001109(n+1). - Creighton Dement, Sep 16 2004

a(n) = (1/2) + (1/4)(3+2*sqrt(2))^n + (1/4)(3-2*sqrt(2))^n. - Antonio Alberto Olivares, Feb 21 2006

a(n) = A001653(n)-A001652(n); e.g., 50=169-119. - Charlie Marion, Apr 10 2006

a(n) = 2*A001653(m)*A011900(n-m-1) +A002315(m)*A001652(n-m-1) - A001108(m) with m<n; otherwise, a(n) = 2*A001653(m)*A011900(m-n) - A002315(m)*A046090(m-n) - A001108(m). See Link to Generalized Proof re Square Triangular Numbers. - Kenneth J Ramsey, Oct 13 2011

a(n) = +7*a(n-1) -7*a(n-2) +1*a(n-3). - Joerg Arndt, Mar 06 2013

a(n) * a(n+2) = (A001108(n)-A001652(n)+3*A046090(n))^2. - Robert Israel, Jul 23 2015

sqrt(a(n+1)*a(n-1)) = a(n)+1 - Bradley Klee & Bill Gosper, Jul 25 2015

a(n) = 1 + sum{k=0..n-1} A002315(k). - David Pasino, Jul 09 2016

E.g.f.: (2*exp(x) + exp((3-2*sqrt(2))*x) + exp((3+2*sqrt(2))*x))/4. - Ilya Gutkovskiy, Jul 09 2016

sqrt(a(n)*(a(n)-1)/2) = A001542(n)/2. - David Pasino, Jul 09 2016

MAPLE

A:= gfun:-rectoproc({a(n) = 6*a(n-1)-a(n-2)-2, a(1) = 1, a(2) = 2}, a(n), remember):

map(A, [$1..100]); # Robert Israel, Jul 22 2015

MATHEMATICA

Table[ 1/4*(2 + (3 - 2*Sqrt[2])^k + (3 + 2*Sqrt[2])^k ) // Simplify, {k, 0, 20}] (* Jean-Fran├žois Alcover, Mar 06 2013 *)

CoefficientList[Series[(1 - 5 x + 2 x^2) / ((1 - x) (1 - 6 x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)

(1 + ChebyshevT[#, 3])/2 & /@ Range[0, 20] (* Bill Gosper, Jul 20 2015 *)

a[1]=1; a[2]=2; a[n_]:=(a[n-1]+1)^2/a[n-2]; a/@Range[25] (* Bradley Klee, Jul 25 2015 *)

LinearRecurrence[{7, -7, 1}, {1, 2, 9}, 30] (* Harvey P. Dale, Dec 06 2015 *)

PROG

(PARI) Vec((1-5*x+2*x^2)/((1-x)*(1-6*x+x^2))+O(x^66)) /* Joerg Arndt, Mar 06 2013 */

(MAGMA) I:=[1, 2, 9]; [n le 3 select I[n] else 7*Self(n-1)-7*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015

CROSSREFS

Cf. A007913, A001541.

A001109(n-1) = sqrt{[(a(n))^2 - (a(n))]/2}.

a(n) = A001108(n-1)+1.

A001110(n-1)=a(n)*(a(n)-1)/2.

Cf. A001652, A001653, A046090.

Identical to A115599, but with additional leading term.

Sequence in context: A138416 A274066 * A115599 A047069 A225006 A211789

Adjacent sequences:  A055994 A055995 A055996 * A055998 A055999 A056000

KEYWORD

easy,nice,nonn

AUTHOR

Barry E. Williams, Jun 14 2000

STATUS

approved

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Last modified February 23 00:33 EST 2018. Contains 299473 sequences. (Running on oeis4.)