A271105 Square 1000-gonal numbers (or square chiliagonal numbers): numbers that are square and chiliagonal (or 1000-gonal).

1, 2504902401, 168859192076889601, 241636344867909601, 16289064572957666645861601, 1098070014289567941239426235218401, 1571330653655890087598658185258401, 105925731068562297456560368093353713060001, 7140610715067574113911463073574478824869628906401

Offset: 1

Author

Muniru A Asiru, Mar 30 2016

Keywords

• nonn
• easy

Comments

a(n) is a number that is a square and a chiliagon. A chiliagon is a polygon with 1000 sides.

Each a(n) ends with digit 1. The remainder of the division of a(n) by 5 is 1.

The remainder of the division of a(n) by 9 is the periodic sequence: 1, 0, 4, 7, 0, 7, 4, 0, 1 of period 9. - Muniru A Asiru, Apr 10 2016

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., 3844891281 = sigma(a(2)) < 2*a(2) = 5009804802. - 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

			2504902401 is in the sequence because 50049^2 = 2504902401 and the 2241th 1000-gonal number is 2504902401. - _Colin Barker_, Mar 31 2016
		

Links

• Colin Barker, Table of n, a(n) for n = 1..190
• M. A. Asiru, All square chiliagonal numbers, Int J Math Edu Sci Technol, 47:7(2016), 1123-1134.
• Index entries for linear recurrences with constant coefficients, signature (1,0,6502873789598402,-6502873789598402,0,-1,1).

Crossrefs

Cf. A000290 (square), A195163 (1000-gonal).

Programs

GAP

g:=1000; Q0:=(g-4)^2; D1:=2*g-4;
S:=[
2*[ 500, 1 ],
4*[ 1022201, 22880 ],
498*[ 8980, 201 ],
996*[ 1, 0 ],
-2*[- 500, 1 ],
-4*[- 1022201, 22880 ]];;      Length(S);
S1:=Filtered(S,i->IsInt((i[1]+g-4)/(2*g-4)));; Length(S1);  #3
S2:=Filtered([1..Length(S)],i->IsInt((S[i][1]+g-4)/(2*g-4)));; Length(S2);  #3  [ 1, 3, 5 ]
S3:=List(S2,i->S[i]);; Length(S3); #3
u:=40320199;;   v:=902490;;   G:=[[u,2*(g-2)*v],[v,u]];;
A:=List([1..Length(S3)],s->List(List([0..11],i->G^i*TransposedMat([S3[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] );; Length(D2);
D3:=Filtered(D2,i->IsInt(i[1]));;  Length(D3);
D4:=List(D3,i->i[2]^2);;  Length(D4);
D5:=Set(D4);;  Length(D5);

Mathematica

Rest@ CoefficientList[Series[x (1 + 2504902400 x + 168859189571987200 x^2 + 66274279001421598 x^3 + 168859189571987200 x^4 + 2504902400 x^5 + x^6)/((1 - x) (1 - 6502873789598402 x^3 + x^6)), {x, 0, 8}], x] (* Michael De Vlieger, Mar 31 2016 *)

PARI

Vec(x*(1 [phone]*x +168859189571987200*x^2 +66274279001421598*x^3 +168859189571987200*x^4 [phone]*x^5+x^6) / ((1 -x)*(1 -6502873789598402*x^3 +x^6)) + O(x^10)) \\ Colin Barker, Mar 31 2016

Formula

G.f.: x*(1 [phone]*x +168859189571987200*x^2 +66274279001421598*x^3 +168859189571987200*x^4 [phone]*x^5+x^6) / ((1 -x)*(1 -6502873789598402*x^3 +x^6)). - Colin Barker, Mar 31 2016

a(n) = A271470(n)*(499*A271470(n)-498). - Muniru A Asiru, Apr 10 2016

a(n) = (A271115(n))^2. - Muniru A Asiru, Apr 10 2016

Extensions

More terms from Colin Barker, Mar 31 2016