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A129556 Numbers n such that centered pentagonal number A005891(n) = (5n^2+5n+2)/2 is a perfect square. 11
0, 2, 21, 95, 816, 3626, 31005, 137711, 1177392, 5229410, 44709909, 198579887, 1697799168, 7540806314, 64471658493, 286352060063, 2448225223584, 10873837476098, 92968086837717, 412919472031679, 3530339074609680, 15680066099727722, 134059916748330141 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Corresponding numbers k>0 such that k^2 is a centered pentagonal number are listed in A129557(n) = {1, 4, 34, 151, 1291, 5734, 49024, ...}.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Centered Pentagonal Number

Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1).

FORMULA

For n>=5, a(n) = 38*a(n-2) - a(n-4) + 18. - Max Alekseyev, May 08 2009

G.f.: x^2*(x^3+2*x^2-19*x-2) / ((x-1)*(x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Feb 21 2013

a(n) = (A221874(n) - 1) / 2. - Bruno Berselli, Feb 21 2013

MAPLE

A005891 := proc(n) (5*n^2+5*n+2)/2 ; end: n := 0 : while true do if issqr(A005891(n)) then print(n) ; fi ; n := n+1 ; od : # R. J. Mathar, Jun 06 2007

MATHEMATICA

Do[ f=(5n^2+5n+2)/2; If[ IntegerQ[ Sqrt[f] ], Print[n] ], {n, 1, 40000} ]

LinearRecurrence[{1, 38, -38, -1, 1}, {0, 2, 21, 95, 816}, 30] (* Harvey P. Dale, Nov 09 2017 *)

PROG

(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 1, -1, -38, 38, 1]^(n-1)*[0; 2; 21; 95; 816])[1, 1] \\ Charles R Greathouse IV, Feb 11 2019

CROSSREFS

Cf. A005891 (centered pentagonal numbers), A129557 (numbers k>0 such that k^2 is a centered pentagonal number), A221874.

Cf. numbers m such that k*A000217(m)+1 is a square: A006451 for k=1; m=0 for k=2; A233450 for k=3; A001652 for k=4; this sequence for k=5; A001921 for k=6. - Bruno Berselli, Dec 16 2013

Sequence in context: A034520 A111128 A213827 * A077209 A068045 A188530

Adjacent sequences:  A129553 A129554 A129555 * A129557 A129558 A129559

KEYWORD

nonn,easy

AUTHOR

Alexander Adamchuk, Apr 20 2007

EXTENSIONS

More terms from R. J. Mathar, Jun 06 2007

Formula and further terms from Max Alekseyev, May 08 2009

a(22)-a(23) from Colin Barker, Feb 21 2013

STATUS

approved

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Last modified November 13 10:29 EST 2019. Contains 329093 sequences. (Running on oeis4.)