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A271470 a(n)-th chiliagonal (or 1000-gonal) number is square. 4

%I

%S 1,2241,18395521,22005481,180674890281,1483422094617961,

%T 1774530705782041,14569695060825930201,119623748111985974353561,

%U 143098862377484625247441,1174906008443637039413730321,9646506658002296058866816899921,11539549215467584644303744700081

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

%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., 18399811 = sigma(a(3)) < 2*a(3) = 36791042. - _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="/A271470/b271470.txt">Table of n, a(n) for n = 1..380</a>

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,80640398,-80640398,0,-1,1).

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

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

%F a(n) = 40320199*a(n-3) + 1804980*A271115(n-3) - 20119698, for n>3. - _Muniru A Asiru_, Apr 09 2016

%F 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

%e a(2)=2241.

%e The 2241st chiliagonal number is a square because 2241*(499*2241 - 498) = 2504902401 = (A271115(2))^2 = A271105(2);

%e the 22005481st chiliagonal number is a square because 22005481*(499*22005481 - 498) = (A271115(4))^2 = A271105(4).

%o (GAP)

%o g:=1000;

%o S:=[2*[ 500, 1 ], 4*[ 1022201, 22880 ], 498*[ 8980, 201 ], 996*[ 1, 0 ],-2*[- 500, 1 ], -4*[- 1022201, 22880 ]];; Length(S);

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

%o A:=List([1..Length(S)],s->List(List([0..6],i->G^i*TransposedMat([S[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] );;

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

%o D4:=Filtered(D3,i->i[2]>0);

%o D5:=List(D4,i->i[1]); # chiliagonal (or 1000-gonal) number is square

%o (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

%Y Cf. A271115, A271105.

%K nonn,easy

%O 1,2

%A _Muniru A Asiru_, Apr 08 2016

%E More terms from _Colin Barker_, Apr 09 2016

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Last modified October 18 22:26 EDT 2019. Contains 328211 sequences. (Running on oeis4.)