A195163 -id:A195163 - cp's OEIS frontend

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

Muniru A Asiru, Mar 30 2016

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

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

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).

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

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);

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 *)

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

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
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

More terms from Colin Barker, Mar 31 2016

A271115 Numbers whose square is a chiliagonal (or 1000-gonal) number.

Original entry on oeis.org

1, 50049, 410924801, 491565199, 4035971329551, 33137139500710799, 39640013290309201, 325462334331581751249, 2672192117918839703333201, 3196586448455823020136799, 26245412174507812354285027551, 215486635921438132237851754543199

Offset: 1

Muniru A Asiru, Mar 31 2016

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

			50049 is in the sequence because 50049^2 = 2504902401, which is the 2241st 1000-gonal number. - _Colin Barker_, Apr 01 2016

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).

Cf. A000290, A195163, A271105.

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)

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 *)
LinearRecurrence[{0,0,80640398,0,0,-1},{1,50049,410924801,491565199,4035971329551,33137139500710799},20] (* Harvey P. Dale, Sep 01 2025 *)

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 */

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

Merged with identical sequence submitted by Colin Barker, Apr 01 2016. - N. J. A. Sloane, Apr 06 2016

A261276 100-gonal numbers: a(n) = 98n(n-1)/2 + n.

Original entry on oeis.org

0, 1, 100, 297, 592, 985, 1476, 2065, 2752, 3537, 4420, 5401, 6480, 7657, 8932, 10305, 11776, 13345, 15012, 16777, 18640, 20601, 22660, 24817, 27072, 29425, 31876, 34425, 37072, 39817, 42660, 45601, 48640, 51777, 55012, 58345, 61776, 65305, 68932, 72657, 76480

Offset: 0

Sergey Pavlov, Aug 13 2015

According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 100).

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A167149, A195163, A249911.

A261276:=List([0..10^2],n->(98*n*(n-1))/2 + n); # Muniru A Asiru, Sep 27 2017

JavaScript
```
function a(n){return 98*n*(n-1)/2+n}
```

A261276:=seq((98*n*(n-1))/2 + n,n=0..10^2); # Muniru A Asiru, Sep 27 2017

Table[n (49 n - 48), {n, 0, 40}] (* Bruno Berselli, Aug 20 2015 *)
PolygonalNumber[100,Range[0,40]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{0,1,100},50] (* Harvey P. Dale, Jan 04 2019 *)

first(m)=vector(m,i,i--;98*i*(i-1)/2 + i) \\ Anders Hellström, Aug 20 2015

a(n) = n*(49*n - 48).
G.f.: x*(1+97*x)/(1-x)^3. [Bruno Berselli, Aug 20 2015]
E.g.f.: exp(x)*(x + 49*x^2). - Nikolaos Pantelidis, Feb 12 2023

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

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A271115 Numbers whose square is a chiliagonal (or 1000-gonal) number.

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A261276 100-gonal numbers: a(n) = 98*n*(n-1)/2 + n.

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Mathematica

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A261276 100-gonal numbers: a(n) = 98n(n-1)/2 + n.