A121351 a(n) = (3*n+1)*a(n-1) - a(n-2), starting a(0)=0, a(1)=1.

0, 1, 7, 69, 890, 14171, 268359, 5889727, 146974816, 4109405121, 127244583935, 4322206448669, 159794394016818, 6387453554224051, 274500708437617375, 12620645134576175199, 618137110885794967376, 32130509120926762128353

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

3*a(n)= Pi*BesselJ_{4/3 + n}(2/3)* BesselY_{4/3}(2/3) - Pi*BesselJ_{4/3}(2/3) * BesselY_{4/3 + n}(2/3).

a(n) = sum {k = 0..floor((n-1)/2)} (-1)^k*3^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k+1/3,k+4/3), cf. A058798. - Peter Bala, Aug 01 2013

a(n) ~ n! * BesselJ(4/3, 2/3) * 3^(n+1/3) * n^(1/3). - Vaclav Kotesovec, Jul 31 2014

Mathematica

f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == (3*n + 1)*a[n - 1] - a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] Rationalize[N[Table[f[n], {n, 0, 25}], 100], 0]
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(3n+1)*a[n-1]-a[n-2]}, a, {n, 20}]  (* Vaclav Kotesovec, Jul 31 2014 *)
nxt[{n_, a_, b_}]:={n+1, b, (3n+4)b-a}; NestList[nxt, {1, 0, 1}, 20][[All, 2]] (* Harvey P. Dale, Jun 20 2021 *)

Crossrefs

Cf. A053984, A058798, A121323, A121353.

Sequence in context: A243668 A265033 A226270 * A302353 A059321 A217400

Adjacent sequences: A121348 A121349 A121350 * A121352 A121353 A121354

Keyword

nonn,easy

Author

Roger L. Bagula and Bob Hanlon (hanlonr(AT)cox.net), Sep 05 2006

Status

approved