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A129444 Numbers n such that centered triangular number A005448(n) = 3n(n-1)/2 + 1 is a perfect square. 9
0, 1, 2, 7, 16, 65, 154, 639, 1520, 6321, 15042, 62567, 148896, 619345, 1473914, 6130879, 14590240, 60689441, 144428482, 600763527, 1429694576, 5946945825, 14152517274, 58868694719, 140095478160, 582740001361, 1386802264322 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Corresponding numbers k>0 such that k^2 is a centered triangular number are listed in A129445(n) = {1, 2, 8, 19, 79, 188, 782, 1861, 7741, 18422, 76628, 182359, ...}.

LINKS

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-1,1).

FORMULA

a(n) = 1/2 + sqrt[ 1/4 + 2/3*(A129445(n)^2 - 1) ].

a(1)=0; a(2)=1; a(3)=2; a(4)=7; a(5)=16; a(6)=65; a(n)=11(a(n-2)-a(n-4))+a(n-6). - Zak Seidov, Apr 17 2007

a(n) = 1 - a(-n+3) for all n in Z. - Michael Somos, Apr 05 2008

G.f.: (x^2 + x^3 - 5*x^4 - x^5) / ((1 - x) * (1 - 10*x^2 + x^4)). - Michael Somos, Apr 05 2008

a(1)=0, a(2)=1, a(3)=2, a(4)=7, a(5)=16, a(n)=a(n-1)+10*a(n-2)- 10*a(n-3)- a(n-4)+a(n-5). - Harvey P. Dale, Dec 06 2012

EXAMPLE

G.f. = x^2 + 2*x^3 + 7*x^4 + 16*x^5 + 65*x^6 + 154*x^7 + 639*x^8 + 1520*x^9 + ...

MATHEMATICA

Do[ f = 3n(n-1)/2 + 1; If[ IntegerQ[ Sqrt[f] ], Print[ n ] ], {n, 1, 150000} ]

a[1]=0; a[2]=1; a[3]=2; a[4]=7; a[5]=16; a[6]=65; a[n_]:=a[n]=11(a[n-2]-a[n-4])+a[n-6]; Table[a[n], {n, 100}] (* Zak Seidov, Apr 17 2007 *)

LinearRecurrence[{1, 10, -10, -1, 1}, {0, 1, 2, 7, 16}, 30] (* Harvey P. Dale, Dec 06 2012 *)

PROG

(PARI) {a(n) = my(m); m = if( n<1, 2-n, n-1); (n<1) + (-1)^(n<1) * polcoeff( (x + x^2 - 5*x^3 - x^4) / ((1 - x) * (1 - 10*x^2 + x^4)) + x * O(x^m), m)}; /* Michael Somos, Apr 05 2008 */

CROSSREFS

Cf. A005448 = Centered triangular numbers: 3n(n-1)/2 + 1. Cf. A129445 = numbers k>0 such that k^2 is a centered triangular number.

Cf. A000290, A249483.

Sequence in context: A239425 A042689 A073998 * A079815 A325510 A006883

Adjacent sequences:  A129441 A129442 A129443 * A129445 A129446 A129447

KEYWORD

nonn,easy

AUTHOR

Alexander Adamchuk, Apr 15 2007

EXTENSIONS

More terms from Zak Seidov, Apr 17 2007

STATUS

approved

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Last modified April 19 00:03 EDT 2021. Contains 343098 sequences. (Running on oeis4.)