Original entry on oeis.org
1, 4, 19, 48, 645, 5346, 9989, 423680, 4936673, 22863284, 717864203, 10398234146, 14778845999, 2318706892436, 41349958502663, 67290481692176, 1273710986008283, 21639017114636720, 1870679510063123381
Offset: 0
a(16) = 1 + 3 + 15 + 29 + 597 + 4701 + 4643 + 413691 + 4512993 + 17926611 + 695000919 + 9680369943 + 4380611853 + 2303928046437 + 39031251610227 + 25940523189513 + 1206420504316107 = 1273710986008283 is prime.
A053518
Numerators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).
Original entry on oeis.org
1, 5, 23, 45, 925, 7285, 7195, 641075, 6993545, 27779915, 1077005935, 15001154095, 6788401045, 3570274674605, 60484653310955, 40198648188145, 1869525647793155, 31559031031400605, 2865359642850975565
Offset: 0
Convergents are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...
- L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
- E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.
- M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.
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for j from 1 to 50 do printf(`%d,`,numer(cfrac([1,seq([i,i+1],i=2..j)]))); od:
-
num[0]=1; num[1]=5; num[n_] := num[n] = (n+2)*num[n-1] + (n+1)*num[n-2]; den[0]=1; den[1]=3; den[n_] := den[n] = (n+2)*den[n-1] + (n+1)*den[n-2]; a[n_] := Numerator[num[n]/den[n]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 16 2013 *)
A053520
Denominators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).
Original entry on oeis.org
2, 8, 38, 74, 1522, 11986, 11838, 1054766, 11506538, 45706526, 1772006854, 24681524038, 11169012898, 5874202721042, 99515904921182, 66139171377658, 3075946152109262, 51924337160029042, 4714400135799462226
Offset: 0
Convergents are 1/2, 3/8, 15/38, 29/74, 597/1522, 4701/11986, ...
-
for j from 1 to 50 do printf(`%d,`,denom(cfrac([0,seq([i,i+1],i=1..j)]))); od:
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a[n_] := ContinuedFractionK[k, k+1, {k, n+1}] // Denominator; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 21 2016 *)
Showing 1-3 of 3 results.
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