A254710 -id:A254710 - cp's OEIS frontend

A254709 Indices of pentagonal numbers (A000326) which are also centered square numbers (A001844).

1, 2, 10, 25, 137, 346, 1906, 4817, 26545, 67090, 369722, 934441, 5149561, 13015082, 71724130, 181276705, 998988257, 2524858786, 13914111466, 35166746297, 193798572265, 489809589370, 2699265900242, 6822167504881, 37595924031121, 95020535478962

Offset: 1

Colin Barker, Feb 06 2015

Also positive integers x in the solutions to 3*x^2 - 4*y^2 - x + 4*y - 2 = 0, the corresponding values of y being A254710.

			10 is in the sequence because the 10th pentagonal number is 145, which is also the 9th centered square number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-1,1).

Cf. A000326, A001844, A254710, A254711.

LinearRecurrence[{1,14,-14,-1,1},{1,2,10,25,137},30] (* Harvey P. Dale, Aug 08 2017 *)

Vec(-x*(x^4+x^3-6*x^2+x+1)/((x-1)*(x^2-4*x+1)*(x^2+4*x+1)) + O(x^100))

a(n) = a(n-1)+14*a(n-2)-14*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+x^3-6*x^2+x+1) / ((x-1)*(x^2-4*x+1)*(x^2+4*x+1)).

A254711 Pentagonal numbers (A000326) which are also centered square numbers (A001844).

Original entry on oeis.org

1, 5, 145, 925, 28085, 179401, 5448301, 34802825, 1056942265, 6751568605, 205041351065, 1309769506501, 39776965164301, 254088532692545, 7716526200523285, 49291865572847185, 1496966305936352945, 9562367832599661301, 290403746825451948001

Offset: 1

Colin Barker, Feb 06 2015

			145 is in the sequence because it is the 10th pentagonal number and the 9th centered square number.

Colin Barker, Table of n, a(n) for n = 1..875
Index entries for linear recurrences with constant coefficients, signature (1,194,-194,-1,1).

Cf. A000326, A001844, A254709, A254710.

Vec(-x*(x^4+4*x^3-54*x^2+4*x+1)/((x-1)*(x^2-14*x+1)*(x^2+14*x+1)) + O(x^100))

a(n) = a(n-1)+194*a(n-2)-194*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+4*x^3-54*x^2+4*x+1) / ((x-1)*(x^2-14*x+1)*(x^2+14*x+1)).

cp's OEIS Frontend

A254709 Indices of pentagonal numbers (A000326) which are also centered square numbers (A001844).

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