A036411 -id:A036411 - cp's OEIS frontend

A048912 Duplicate of A036411.

1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641

Offset: 1

dead

A342709 12-gonal (dodecagonal) square numbers.

1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, 7442093853169599697984, 349619996931001511354641, 16424697761903901433970161

Offset: 1

Bernard Schott, Mar 19 2021

The 12-gonal square numbers k correspond to the nonnegative integer solutions of the Diophantine equation k = d*(5*d-4) = c^2, equivalent to (5*d-2)^2 - 5*c^2 = 4. Substituting x = 5*d-2 and y = c gives the Pell-Fermat's equation x^2 - 5*y^2 = 4.
The solutions x are in A342710, while corresponding solutions y that are also the indices c of the squares which are 12-gonal are in A033890.
The indices d of the corresponding 12-gonal which are squares are in A081068.

			142129 = 169*(5*169-4) = 377^2, so 142129 is the 169th 12-gonal number and the 377th square, hence 142129 is a term.

Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Intersection of A000290 (squares) and A051624 (12-gonal numbers).

Similar for n-gonal squares: A001110 (triangular), A036353 (pentagonal), A046177 (hexagonal), A036354 (heptagonal), A036428 (octagonal), A036411 (9-gonal), A188896 (there are no 10-gonal squares > 1), A333641 (11-gonal), this sequence (12-gonal).

with(combinat):
seq(fibonacci(4*n-2)^2, n=1..16);

Table[Fibonacci[4*n - 2]^2, {n, 1, 16}] (* Amiram Eldar, Mar 19 2021 *)

a(n) = fibonacci(4*n-2)^2; \\ Michel Marcus, Mar 21 2021

G.f.: x*(1 + 16*x + x^2)/((1 - x)*(1 - 47*x + x^2)). - Stefano Spezia, Mar 20 2021
a(n) = 48*a(n-1) - 48*a(n-2) + a(n-3). - Kevin Ryde, Mar 20 2021
a(n) = 9*A161582(n) + 1. - Hugo Pfoertner, Mar 19 2021
a(n) = A033890(n-1)^2.

A048911 Indices of square numbers which are also 9-gonal.

Original entry on oeis.org

1, 3, 33, 91, 989, 2727, 29637, 81719, 888121, 2448843, 26613993, 73383571, 797531669, 2199058287, 23899336077, 65898365039, 716182550641, 1974751892883, 21461577183153, 59176658421451, 643131132943949, 1773325000750647, 19272472411135317, 53140573364097959

Offset: 1

Eric W. Weisstein

From Ant King, Nov 18 2011: (Start)
lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)).
lim( n -> Infinity, a(2n)/a(2n-1)) = 1/25 * (39 + 8 * sqrt(14)).
(End)

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Nonagonal Square Number.
Index entries for linear recurrences with constant coefficients, signature (0,30,0,-1)

Cf. A048910, A036411.

LinearRecurrence[ {0, 30, 0, - 1 }, { 1, 3, 33, 91 } , 21 ] (* Ant King, Nov 18 2011 *)

Vec(x*(x+1)^3/(x^4-30*x^2+1) + O(x^50)) \\ Colin Barker, Jun 22 2015

From Ant King, Nov 18 2011: (Start)
a(n) = 30 * a(n-2) - a(n-4).
G.f.: x * (1 + x) ^ 3 / (1 - 30 * x ^ 2 + x ^ 4).
Let p = 8 * sqrt(7) + 9 * sqrt(14) - 7 * sqrt(2) - 28 and q = 7 * sqrt(2) + 9 * sqrt(14) - 8 * sqrt(7) - 28. Then
a(n) = 1/112 * ( ( p + q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7)) ^ n - ( p - q * (-1) ^ n) * ( 2 * sqrt(2) - sqrt(7)) ^ ( n - 1) ).
a(n) = floor ( 1/112 * ( p + q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7)) ^ n ).
(End)

A048910 Indices of 9-gonal numbers that are also square.

Original entry on oeis.org

1, 2, 18, 49, 529, 1458, 15842, 43681, 474721, 1308962, 14225778, 39225169, 426298609, 1175446098, 12774732482, 35224157761, 382815675841, 1055549286722, 11471695542738, 31631254443889, 343768050606289, 947882084029938, 10301569822645922, 28404831266454241

Offset: 1

Eric W. Weisstein

From Ant King, Nov 18 2011: (Start)
lim( n -> Infinity, a(2n+1)/a(2n)) = 1/25 * (137 + 36 * sqrt(14)) = 1/25 * (9 + 2 * sqrt(14))^2.
lim( n -> Infinity, a(2n)/a(2n-1)) = 1/25 * (39 + 8 * sqrt(14)).
(14 * a(n) - 5)^2 - 56 *  A048911(n) ^ 2 = 25.
(End)

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Nonagonal Square Number
Index entries for linear recurrences with constant coefficients, signature (1,30,-30,-1,1).

Cf. A048911, A036411.

LinearRecurrence[ {1, 30, - 30, -1, 1 }, {1, 2, 18, 49, 529}, 21 ] (* Ant King, Nov 18 2011 *)

Vec(-x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^4-30*x^2+1)) + O(x^50)) \\ Colin Barker, Jun 22 2015

From Ant King, Nov 18 2011: (Start)
a(n) = 30 * a(n - 2) - a(n-4) - 10.
a(n) = a(n - 1) + 30 * a(n - 2) - 30 * a(n - 3) - a(n - 4) + a(n - 5).
Let p = 9 + 4 * sqrt(2) + sqrt(7) + 2 * sqrt(14) and q = 9 - 4 * sqrt(2) - sqrt(7) + 2 * sqrt(14). Then
a(n) = 1/56 * ( ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) + ( p + q * (-1)^n) * ( 2 * sqrt(2) - sqrt(7))^n + 20 ).
a(n) = ceiling (1/56 * ( p - q * (-1) ^ n) * ( 2 * sqrt(2) + sqrt(7))^(n - 1) ).
G.f.: x * (1 + x - 14 * x^2 + x^3 + x^4) / ((1 - x) * (1 - 30 * x^2 + x^4)).
(End)

A333641 11-gonal (or hendecagonal) square numbers.

Original entry on oeis.org

0, 1, 196, 29241, 1755625, 261468900, 38941102225, 2337990844401, 348201795147556, 51858411008887561, 3113535139359330841, 463705205422871375236, 69060571958250748760481, 4146338334574433921200225, 617522713934165528806340100, 91968930524758079223806760025

Offset: 1

Bernard Schott, Mar 31 2020

The 11-gonal square numbers correspond to the nonnegative integer solutions of the Diophantine equation k*(9*k-7)/2 = m^2, equivalent to (18*k-7)^2 - 72*m^2 = 49. Substituting x = 18*k-7 and y = m gives the Pell equation x^2-72*y^2 = 49. The integer solutions (x,y) = (-7,0), (11,1), (119,14), (1451,171), (11243,1325), ... correspond to the following solutions (k,m) = (0,0), (1,1), (7,14), (81,171), (625,1325), ...

			1755625 is a term because 625*(9*625-7)/2 = 1325^2 = 1755625; that means that 1755625 is the 625th 11-gonal number and the square of 1325.

Index entries for linear recurrences with constant coefficients, signature (1,0,1331714,-1331714,0,-1,1).

Intersection of A000290 (squares) and A051682 (11-gonals).
Cf. A106525.
Cf. A001110 (square triangulars), A036353 (square pentagonals), A046177 (square hexagonals), A036354 (square heptagonals), A036428 (square octagonals), A036411 (square 9-gonals), A188896 (only {0,1} are square 10-gonals), this sequence (square 11-gonals), A342709 (square 12-gonals).

for k from 0 to 8000000 do
d:= k*(9*k-7)/2;
if issqr(d) then print(k,sqrt(d),d); else fi; od:

Last /@ Solve[(18*x - 7)^2 - 72*y^2 == 49 && x >= 0 && y >= 0 && y < 10^16, {x, y}, Integers] /. Rule -> (#2^2 &) (* Amiram Eldar, Mar 31 2020 *)

concat(0, Vec(-x*(1 + 195*x + 29045*x^2 + 394670*x^3 + 29045*x^4 + 195*x^5 + x^6)/(-1 + x + 1331714*x^3 - 1331714*x^4 - x^6 + x^7) + O(x^20))) \\ Jinyuan Wang, Mar 31 2020

a(n) = k*(9*k-7)/2 for n > 1, where k = (A106525(4*n-6) + 7)/18. - Jinyuan Wang, Mar 31 2020

More terms from Amiram Eldar, Mar 31 2020

cp's OEIS Frontend

A048912 Duplicate of A036411.

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A342709 12-gonal (dodecagonal) square numbers.

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A048911 Indices of square numbers which are also 9-gonal.

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A048910 Indices of 9-gonal numbers that are also square.

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A333641 11-gonal (or hendecagonal) square numbers.

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