A271470 a(n)-th chiliagonal (or 1000-gonal) number is square.

1, 2241, 18395521, 22005481, 180674890281, 1483422094617961, 1774530705782041, 14569695060825930201, 119623748111985974353561, 143098862377484625247441, 1174906008443637039413730321, 9646506658002296058866816899921, 11539549215467584644303744700081

Offset: 1

Author

Muniru A Asiru, Apr 08 2016

Keywords

• nonn
• easy

Comments

a(n) is odd since a(n) mod 10 = A000012(n). Since all odd numbers with one or two distinct prime factors are deficient, a(n) is deficient. E.g., 18399811 = sigma(a(3)) < 2*a(3) = 36791042. - Muniru A Asiru, Nov 17 2016

The digital root of a(n) is always 1, 4, 7 or 9. - Muniru A Asiru, Nov 29 2016

Examples

			a(2)=2241.
The 2241st chiliagonal number is a square because 2241*(499*2241 - 498) = 2504902401 = (A271115(2))^2 = A271105(2);
the 22005481st chiliagonal number is a square because 22005481*(499*22005481 - 498) = (A271115(4))^2 = A271105(4).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..380
• M. A. Asiru, All square chiliagonal numbers, Int J Math Educ Sci Technol, 47:7(2016), 1123-1134.
• Index entries for linear recurrences with constant coefficients, signature (1,0,80640398,-80640398,0,-1,1).

Crossrefs

Cf. A271115, A271105.

Programs

GAP

g:=1000;
S:=[2*[ 500, 1 ], 4*[ 1022201, 22880 ], 498*[ 8980, 201 ], 996*[ 1, 0 ],-2*[- 500, 1 ], -4*[- 1022201, 22880 ]];;      Length(S);
u:=40320199;;   v:=902490;;   G:=[[u,2*(g-2)*v],[v,u]];;
A:=List([1..Length(S)],s->List(List([0..6],i->G^i*TransposedMat([S[s]])),Concatenation));; Length(A);
D1:=Union(List([1..Length(A)],k->A[k]));; Length(D1);
D2:=List(D1,i-> [(i[1]+(g-4))/(2*(g-2)),i[2]/2] );;
D3:=Filtered(D2,i->IsInt(i[1]));
D4:=Filtered(D3,i->i[2]>0);
D5:=List(D4,i->i[1]); # chiliagonal (or 1000-gonal) number is square

PARI

Vec(x*(1+2240*x+18393280*x^2-77030438*x^3+18393280*x^4+2240*x^5+x^6)/((1-x)*(1-80640398*x^3+x^6)) + O(x^50)) \\ Colin Barker, Apr 09 2016

Formula

a(n)*(499*a(n)-498) = (A271115(n))^2 = A271105(n).

a(n) = 80640398*a(n-3) - a(n-6) - 40239396, for n>6.

a(n) = 40320199*a(n-3) + 1804980*A271115(n-3) - 20119698, for n>3. - Muniru A Asiru, Apr 09 2016

G.f.: x*(1+2240*x+18393280*x^2-77030438*x^3+18393280*x^4+2240*x^5+x^6) / ((1-x)*(1-80640398*x^3+x^6)). - Colin Barker, Apr 09 2016

Extensions

More terms from Colin Barker, Apr 09 2016