OFFSET
1,2
COMMENTS
a(n) is odd since a(n) mod 10 = 1 or 9. Since all odd numbers with one or two distinct prime factors are deficient, a(n) is deficient. E.g., 7294309908480 = sigma(a(5)) < 2*a(5) = 8071942659102. - Muniru A Asiru, Nov 17 2016
The digital root of a(3*n) is A131598(n-1). - Muniru A Asiru, Dec 01 2016
LINKS
Colin Barker, Table of n, a(n) for n = 1..350
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 (0,0,80640398,0,0,-1).
FORMULA
a(n)^2 = A271105(n).
a(n) = 80640398*a(n-3)-a(n-6) for n>6. - Colin Barker, Apr 01 2016
G.f.: x*(1+x)*(1+50048*x+410874753*x^2+50048*x^3+x^4) / (1-80640398*x^3+x^6). - Colin Barker, Apr 01 2016
a(n) = 40320199*a(n-3) + 900685020*A271470(n-3) - 449440020 for n>3. - Muniru A Asiru, Apr 09 2016
A010888(a(3*n)) = A131598(n-1) where A131598 has period 3: repeat [2, 5, 8] and A010888 is digital root. - Michel Marcus, Dec 04 2014
EXAMPLE
50049 is in the sequence because 50049^2 = 2504902401, which is the 2241st 1000-gonal number. - Colin Barker, Apr 01 2016
MATHEMATICA
Rest@ CoefficientList[Series[x (1 + x) (1 + 50048 x + 410874753 x^2 + 50048 x^3 + x^4)/(1 - 80640398 x^3 + x^6), {x, 0, 12}], x] (* Michael De Vlieger, Apr 01 2016 *)
PROG
(PARI) Vec(x*(1+x)*(1+50048*x+410874753*x^2+50048*x^3+x^4)/(1-80640398*x^3+x^6) + O(x^15)) /* Colin Barker, Apr 01 2016 */
(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 ]];;
S1:=Filtered(S, i->IsInt((i[1]+g-4)/(2*g-4)));;
S2:=Filtered([1..Length(S)], i->IsInt((S[i][1]+g-4)/(2*g-4)));;
S3:=List(S2, i->S[i]);;
u:=40320199;; v:=902490;; G:=[[u, 2*(g-2)*v], [v, u]];;
A:=List([1..Length(S3)], s->List(List([0..6], 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]));
D4:=Filtered(D3, i->i[2]>0);
D5:=List(D4, i->i[2]); # indices of square numbers for square 1000 gonal numbers (or square chiliagonal numbers)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Muniru A Asiru, Mar 31 2016
EXTENSIONS
Merged with identical sequence submitted by Colin Barker, Apr 01 2016. - N. J. A. Sloane, Apr 06 2016
STATUS
approved