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Square 1000-gonal numbers (or square chiliagonal numbers): numbers that are square and chiliagonal (or 1000-gonal).
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%I #49 Nov 30 2016 05:32:17

%S 1,2504902401,168859192076889601,241636344867909601,

%T 16289064572957666645861601,1098070014289567941239426235218401,

%U 1571330653655890087598658185258401,105925731068562297456560368093353713060001,7140610715067574113911463073574478824869628906401

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

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

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

%C 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

%C 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

%C The digital root of a(n) is always 1, 4, 7 or 9. - _Muniru A Asiru_, Nov 29 2016

%H Colin Barker, <a href="/A271105/b271105.txt">Table of n, a(n) for n = 1..190</a>

%H M. A. Asiru, <a href="http://dx.doi.org/10.1080/0020739X.2016.1164346">All square chiliagonal numbers</a>, Int J Math Edu Sci Technol, 47:7(2016), 1123-1134.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6502873789598402,-6502873789598402,0,-1,1).

%F G.f.: 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)). - _Colin Barker_, Mar 31 2016

%F a(n) = A271470(n)*(499*A271470(n)-498). - _Muniru A Asiru_, Apr 10 2016

%F a(n) = (A271115(n))^2. - _Muniru A Asiru_, Apr 10 2016

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

%t 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 *)

%o (GAP)

%o g:=1000; Q0:=(g-4)^2; D1:=2*g-4;

%o S:=[

%o 2*[ 500, 1 ],

%o 4*[ 1022201, 22880 ],

%o 498*[ 8980, 201 ],

%o 996*[ 1, 0 ],

%o -2*[- 500, 1 ],

%o -4*[- 1022201, 22880 ]];; Length(S);

%o S1:=Filtered(S,i->IsInt((i[1]+g-4)/(2*g-4)));; Length(S1); #3

%o S2:=Filtered([1..Length(S)],i->IsInt((S[i][1]+g-4)/(2*g-4)));; Length(S2); #3 [ 1, 3, 5 ]

%o S3:=List(S2,i->S[i]);; Length(S3); #3

%o u:=40320199;; v:=902490;; G:=[[u,2*(g-2)*v],[v,u]];;

%o A:=List([1..Length(S3)],s->List(List([0..11],i->G^i*TransposedMat([S3[s]])),Concatenation));; Length(A);

%o D1:=Union(List([1..Length(A)],k->A[k]));; Length(D1);

%o D2:=List(D1,i-> [(i[1]+(g-4))/(2*(g-2)),i[2]/2] );; Length(D2);

%o D3:=Filtered(D2,i->IsInt(i[1]));; Length(D3);

%o D4:=List(D3,i->i[2]^2);; Length(D4);

%o D5:=Set(D4);; Length(D5);

%o (PARI) Vec(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)) + O(x^10)) \\ _Colin Barker_, Mar 31 2016

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

%K nonn,easy

%O 1,2

%A _Muniru A Asiru_, Mar 30 2016

%E More terms from _Colin Barker_, Mar 31 2016