A254137 -id:A254137 - cp's OEIS frontend

A254136 Indices of pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

1, 73, 889, 84049, 1025713, 96992281, 1183671721, 111929008033, 1365956140129, 129165978277609, 1576312202036953, 149057427003352561, 1819062915194503441, 172012141595890577593, 2099197027822254933769, 198501862344230723189569, 2422471551043966999065793

Offset: 1

Colin Barker, Jan 26 2015

Also positive integers x in the solutions to 3*x^2 - 6*y^2 - x + 6*y - 2 = 0, the corresponding values of y being A254137.

			73 is in the sequence because the 73rd pentagonal number is 7957, which is also the 52nd centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..653
Index entries for linear recurrences with constant coefficients, signature (1,1154,-1154,-1,1).

Cf. A000326, A003215, A254137, A254138.

LinearRecurrence[{1,1154,-1154,-1,1},{1,73,889,84049,1025713},20] (* Harvey P. Dale, Mar 24 2024 *)

Vec(-x*(x^4+72*x^3-338*x^2+72*x+1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+72*x^3-338*x^2+72*x+1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).

A254138 Pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 7957, 1185037, 10596309577, 1578130224697, 14111253811878301, 2101618114050816901, 18792154258821103289617, 2798754265133491448134897, 25025774916617575492416996517, 3727140237435880812247465267837, 33327174817289665775049786996211801

Offset: 1

Colin Barker, Jan 26 2015

			7957 is in the sequence because it is the 73rd pentagonal number and the 52nd centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..327
Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).

Cf. A000326, A003215, A254136, A254137.

LinearRecurrence[{1,1331714,-1331714,-1,1},{1,7957,1185037,10596309577,1578130224697},20] (* Harvey P. Dale, Sep 26 2023 *)

Vec(-x*(x^4+7956*x^3-154634*x^2+7956*x+1)/((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)) + O(x^100))

a(n) = a(n-1)+1331714*a(n-2)-1331714*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+7956*x^3-154634*x^2+7956*x+1) / ((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)).

cp's OEIS Frontend

A254136 Indices of pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

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