A137881 -id:A137881 - cp's OEIS frontend

A137878 Perfect squares among 17-gonal numbers A051869(k) = k(15k - 13)/2.

1, 17689, 378225, 4109707449, 87870152041, 954775454112481, 20414169462254569, 221815343046210267025, 4742660677722035990769, 51532584126226886201833161, 1101824413949324675985344641, 11972153009151467313136073526409, 255978051492792346696545201859225

Offset: 1

Alexander Adamchuk, Feb 19 2008

Corresponding square roots sqrt(a(n)) are listed in A137879.

Indices of perfect squares among the 17-gonal numbers A051869(k) = k*(15*k - 13)/2 are listed in A137880. Note that all such indices are also perfect squares, their square roots are listed in A137881(k) = sqrt(A137880(k)).

Colin Barker, Table of n, a(n) for n = 1..350
Index entries for linear recurrences with constant coefficients, signature (1,232322,-232322,-1,1).

Cf. A051869 (17-gonal numbers), A137879, A137880, A137881.

Vec(x*(1+17688*x+128214*x^2+17688*x^3+x^4)/((1-x)*(1-482*x+x^2)*(1+482*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 19 2016

a(n) = A137879(n)^2 = A051869( A137880(n) ) = A051869( A137881(n)^2 ).
From Colin Barker, Jun 19 2016: (Start)
a(n) = a(n-1) + 232322*a(n-2) - 232322*a(n-3) - a(n-4) + a(n-5) for n > 5.
G.f.: x*(1 + 17688*x + 128214*x^2 + 17688*x^3 + x^4) / ((1-x)*(1 - 482*x + x^2)*(1 + 482*x + x^2)).
(End)

Edited and extended by Max Alekseyev, Oct 19 2008

A137879 Numbers k such that k^2 is a 17-gonal number.

Original entry on oeis.org

1, 133, 615, 64107, 296429, 30899441, 142878163, 14893466455, 68866978137, 7178619931869, 33193740583871, 3460079913694403, 15999314094447685, 1667751339780770377, 7711636199783200299, 803852685694417627311, 3716992648981408096433

Offset: 1

Alexander Adamchuk, Feb 19 2008

Corresponding 17-gonal numbers equal k^2 are listed in A137878.

The 17-gonal numbers A051869(n) = n*(15n - 13)/2 are perfect squares for indices n listed in A137880. Note that all such indices are also perfect squares of numbers listed in A137881.

Harvey P. Dale, Table of n, a(n) for n = 1..745
Index entries for linear recurrences with constant coefficients, signature (0,482,0,-1).

Cf. A051869 (17-gonal numbers), A137878 (17-gonal numbers that are perfect squares), A137880, A137881.

LinearRecurrence[{0,482,0,-1},{1,133,615,64107},20] (* Harvey P. Dale, May 12 2014 *)

is(n)=ispolygonal(n^2,17) \\ Charles R Greathouse IV, Oct 16 2015

a(n) = sqrt(A137878(n)) = sqrt(A051869(A137880(n))) = sqrt(A051869(A137881(n)^2)).
From Max Alekseyev, Oct 19 2008: (Start)
For n>=5, a(n) = 482*a(n-2) - a(n-4).
a(2n) = (-60 + 17*sqrt(30))/120 * (11 + 2*sqrt(30))^(2n) + (-60 - 17*sqrt(30))/120 * (11 - 2*sqrt(30))^(2n).
a(2n+1) = (60 + 17*sqrt(30))/120 * (11 + 2*sqrt(30))^(2n) + (60 - 17*sqrt(30))/120 * (11 - 2*sqrt(30))^(2n). (End)

Extended by Max Alekseyev, Oct 19 2008

A137880 Indices k of perfect squares among 17-gonal numbers A051869(k) = k(15k - 13)/2.

Original entry on oeis.org

1, 49, 225, 23409, 108241, 11282881, 52171729, 5438325025, 25146664929, 2621261378961, 12120640323841, 1263442546333969, 5842123489426225, 608976686071593889, 2815891401263116401, 293525499243961920321, 1357253813285332678849, 141478681658903574000625

Offset: 1

Alexander Adamchuk, Feb 19 2008

Corresponding perfect squares are listed in A137878.

Note that all a(n) are perfect squares themselves, their square roots are listed in A137881.

Matthew House, Table of n, a(n) for n = 1..746
Index entries for linear recurrences with constant coefficients, signature (1,482,-482,-1,1).

Cf. A051869 (17-gonal numbers), A137878 (17-gonal numbers that are perfect squares), A137879, A137881.

Rest@ CoefficientList[Series[x (1 + 48 x - 306 x^2 + 48 x^3 + x^4)/((1 - x) (1 - 22 x + x^2) (1 + 22 x + x^2)), {x, 0, 18}], x] (* Michael De Vlieger, Jun 18 2016 *)

Vec(x*(1+48*x-306*x^2+48*x^3+x^4)/((1-x)*(1-22*x+x^2)*(1+22*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 18 2016

A051869( a(n) ) = A137878(n); a(n) = A137881(n)^2.
From Max Alekseyev, Oct 19 2008: (Start)
a(n) = 482*a(n-2) - a(n-4) - 208.
a(2n) = ( (15 - sqrt(30))/30 * (11 + 2*sqrt(30))^n + (15 + sqrt(30))/30 * (11 - 2*sqrt(30))^n )^2.
a(2n+1) = ( (15 + sqrt(30))/30 * (11 + 2*sqrt(30))^n + (15 - sqrt(30))/30 * (11 - 2*sqrt(30))^n )^2. (End)
a(n) = a(n-1) + 482*a(n-2) - 482*a(n-3) - a(n-4) + a(n-5). - Matthew House, Jun 18 2016
G.f.: x*(1 + 48*x - 306*x^2 + 48*x^3 + x^4) / ((1-x)*(1 - 22*x + x^2)*(1 + 22*x + x^2)). - Colin Barker, Jun 18 2016

Edited and extended by Max Alekseyev, Oct 19 2008

A238245 Positive integers n such that x^2 - 22xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

20, 39, 56, 71, 80, 84, 95, 104, 111, 116, 119, 120, 156, 180, 191, 224, 239, 255, 284, 296, 311, 320, 336, 351, 359, 380, 399, 404, 416, 431, 444, 455, 464, 471, 476, 479, 480, 500, 504, 551, 596, 599, 624, 639, 680, 695, 696, 719, 720, 756, 764, 791, 824

Offset: 1

Colin Barker, Feb 20 2014

nonn

			39 is in the sequence because x^2 - 22xy + y^2 + 39 = 0 has integer solutions, for example (x, y) = (2, 43).

Cf. A157014 (n = 20), A137881 (n = 104), A077422 (n = 120), A133275 (n = 180).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331, A238240.

cp's OEIS Frontend

A137878 Perfect squares among 17-gonal numbers A051869(k) = k*(15*k - 13)/2.

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A137879 Numbers k such that k^2 is a 17-gonal number.

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A137880 Indices k of perfect squares among 17-gonal numbers A051869(k) = k*(15*k - 13)/2.

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Author

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Programs

Mathematica

PARI

Formula

Extensions

A238245 Positive integers n such that x^2 - 22xy + y^2 + n = 0 has integer solutions.

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A137878 Perfect squares among 17-gonal numbers A051869(k) = k(15k - 13)/2.

A137880 Indices k of perfect squares among 17-gonal numbers A051869(k) = k(15k - 13)/2.