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A129444
Numbers k such that the centered triangular number A005448(k) = 3*k*(k-1)/2 + 1 is a perfect square.
10
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
OFFSET
1,3
COMMENTS
Corresponding numbers m > 0 such that m^2 is a centered triangular number are listed in A129445 = {1, 2, 8, 19, 79, 188, 782, 1861, 7741, 18422, 76628, 182359, ...}.
FORMULA
a(n) = 1/2 + sqrt(1/4 + (2/3)*(A129445(n)^2 - 1)).
a(n) = 11*(a(n-2) - a(n-4)) + a(n-6); a(1)=0; a(2)=1; a(3)=2; a(4)=7; a(5)=16; a(6)=65. - 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*(1 + x - 5*x^2 - x^3) / ((1 - x) * (1 - 10*x^2 + x^4)). - Michael Somos, Apr 05 2008
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5); a(1)=0, a(2)=1, a(3)=2, a(4)=7, a(5)=16. - Harvey P. Dale, Dec 06 2012
a(n) = (1/2)*(2*[n=0] + 1 + ((1+(-1)^n)/2)*(31*b(n/2) - 3*b(n/2 + 1)) + ((1-(-1)^n)/2)*(13*b((n-1)/2) - b((n+1)/2))), where b(n)=A004189(n). - G. C. Greubel, Feb 07 2024
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 */
(Magma) I:=[0, 1, 2, 7, 16, 65]; [n le 6 select I[n] else 11*Self(n-2) -11*Self(n-4) +Self(n-6): n in [1..40]]; // G. C. Greubel, Feb 07 2024
(SageMath)
def A129444_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^2*(1+x-5*x^2-x^3)/((1-x)*(1-10*x^2+x^4)) ).list()
a=A129444_list(40); a[1:] # G. C. Greubel, Feb 07 2024
CROSSREFS
Cf. A005448 (centered triangular numbers).
Cf. A129445 (numbers k > 0 such that k^2 is a centered triangular number).
Sequence in context: A239425 A042689 A073998 * A079815 A362760 A325510
KEYWORD
nonn,easy
AUTHOR
Alexander Adamchuk, Apr 15 2007
EXTENSIONS
More terms from Zak Seidov, Apr 17 2007
STATUS
approved