A271105 -id:A271105 - cp's OEIS frontend

A195163 1000-gonal numbers: a(n) = n(499n - 498).

0, 1, 1000, 2997, 5992, 9985, 14976, 20965, 27952, 35937, 44920, 54901, 65880, 77857, 90832, 104805, 119776, 135745, 152712, 170677, 189640, 209601, 230560, 252517, 275472, 299425, 324376, 350325, 377272, 405217, 434160, 464101, 495040, 526977, 559912, 593845, 628776

Offset: 0

Kausthub Gudipati, Sep 10 2011

a(A271470(n)) is a perfect square. In fact, a(A271470(n)) = A271105(n) if the first term of a(n) is 1. - Muniru A Asiru, Apr 10 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. A. Asiru, All square chiliagonal numbers, Int J Math Educ Sci Technol, 47:7(2016), 1123-1134.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000578, A092182.

a:=[0,1,1000];; for n in [4..10^2] do a[n]:=3*a[n-1]-3*a[n-2]+*a[n-3]; od; a;  # Muniru A Asiru, Sep 12 2017

function a(n){return 998*n*(n-1)/2+n} // Sergey Pavlov, Aug 14 2015

[n*(499*n-498): n in [0..45]]; // Vincenzo Librandi, Nov 25 2011

A195163:=n->n*(499*n - 498): seq(A195163(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

LinearRecurrence[{3,-3,1},{0,1,1000},50] (* Vincenzo Librandi, Nov 25 2011 *)
PolygonalNumber[1000,Range[0,40]] (* Harvey P. Dale, Sep 15 2022 *)

a(n)=n*(499*n-498) \\ Charles R Greathouse IV, Sep 11 2011

x='x+O('x^99); concat(0, Vec(x*(1+997*x)/(1-x)^3)) \\ Altug Alkan, Apr 10 2016

a(n) = 998*n*(n-1)/2 + n, according to the common formula for s-gonal numbers, a(n) = (s-2)*n*(n-1)/2 + n. - Sergey Pavlov, Aug 14 2015
G.f.: x*(1+997*x)/(1-x)^3. - R. J. Mathar, Sep 12 2011
E.g.f.: exp(x)*x*(1 + 499*x). - Ilya Gutkovskiy, Apr 10 2016
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. - Muniru A Asiru, Sep 12 2017

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

Original entry on oeis.org

1, 2241, 18395521, 22005481, 180674890281, 1483422094617961, 1774530705782041, 14569695060825930201, 119623748111985974353561, 143098862377484625247441, 1174906008443637039413730321, 9646506658002296058866816899921, 11539549215467584644303744700081

Offset: 1

Muniru A Asiru, Apr 08 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., 18399811 = sigma(a(3)) < 2*a(3) = 36791042. - 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

			a(2)=2241.
The 2241st chiliagonal number is a square because 2241*(499*2241 - 498) = 2504902401 = (A271115(2))^2 = A271105(2);
the 22005481st chiliagonal number is a square because 22005481*(499*22005481 - 498) = (A271115(4))^2 = A271105(4).

Colin Barker, Table of n, a(n) for n = 1..380
M. A. Asiru, All square chiliagonal numbers, Int J Math Educ Sci Technol, 47:7(2016), 1123-1134.
Index entries for linear recurrences with constant coefficients, signature (1,0,80640398,-80640398,0,-1,1).

Cf. A271115, A271105.

g:=1000;
S:=[2*[ 500, 1 ], 4*[ 1022201, 22880 ], 498*[ 8980, 201 ], 996*[ 1, 0 ],-2*[- 500, 1 ], -4*[- 1022201, 22880 ]];;      Length(S);
u:=40320199;;   v:=902490;;   G:=[[u,2*(g-2)*v],[v,u]];;
A:=List([1..Length(S)],s->List(List([0..6],i->G^i*TransposedMat([S[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] );;
D3:=Filtered(D2,i->IsInt(i[1]));
D4:=Filtered(D3,i->i[2]>0);
D5:=List(D4,i->i[1]); # chiliagonal (or 1000-gonal) number is square

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

a(n)*(499*a(n)-498) = (A271115(n))^2 = A271105(n).
a(n) = 80640398*a(n-3) - a(n-6) - 40239396, for n>6.
a(n) = 40320199*a(n-3) + 1804980*A271115(n-3) - 20119698, for n>3. - Muniru A Asiru, Apr 09 2016
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

More terms from Colin Barker, Apr 09 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

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A195163 1000-gonal numbers: a(n) = n*(499*n - 498).

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

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

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A195163 1000-gonal numbers: a(n) = n(499n - 498).