

A178783


Continued fraction for EulerMascheroni constant with convergents 0/1, 1/1, 1/2, 4/7, etc., which lie between the monotonically increasing series given by (sum k=1 to n, 1/k  sum k=n to n^2, 1/k) and the monotonically decreasing series (sum k=1 to n, 1/k  sum k=n to n^21, 1/k) both of which converge to gamma. Thus each p/q in the sequence lies within 1/q^2 of gamma.


0



0, 1, 1, 3, 4, 5, 3, 13, 5, 2, 10, 3, 4, 2, 42, 12, 3, 8, 9, 2, 6, 50, 5, 67, 5, 7, 12, 401, 2, 2, 3, 3, 4, 6, 3, 3, 12, 3, 2, 2, 2, 5, 6
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OFFSET

0,4


COMMENTS

Series derived from def. gamma=lim(sum k=1 to n, 1/k  log(n)) by noting that 2*gamma = 2*sum k=1 to n, 1/k 2*log(n) (ignoring limit) and also gamma = sum k=1 to n^2, 1/k  log(n^2), then gamma = 2*gamma  gamma gets rid of the log term and the series consists of all rational terms. The decreasing series was found by accident. The proofs for both are straightforward. The PARI program uses the first term of the EulerMaclaurin summation and gamma itself for the upper and lower bounds.


LINKS

Table of n, a(n) for n=0..42.


PROG

(PARI) pconv=vector(43); qconv=vector(43); cf=vector(43); fract=vector(43); pconv[1]=0; pconv[2]=1; pconv[3]=1; pconv[4]=4; qconv[1]=1; qconv[2]=1; qconv[3]=2; qconv[4]=7; cf[1]=0; cf[2]=1; cf[3]=1; cf[4]=3; fract[1]=0/1; fract[2]=1/1; fract[3]=1/2; fract[4]=4/7; for(k=5, 43, tst=0; cfm=1; until(tst==1, pp = cfm * pconv[k  1] + pconv[k  2]; pn = cfm * pconv[k  1]  pconv[k  2]; qp = cfm * qconv[k  1] + qconv[k  2]; qn = cfm * qconv[k  1]  qconv[k  2]; slp = pp/qp; sln = pn/qn; if(((Euler  slp < 2/(3 * qp^2) && Euler  slp > 0) 
(slp  Euler < 1/(3 * qp^2) && slp  Euler > 0))  ((Euler  sln < 2/(3 * qn^2) && Euler  sln > 0)  (sln  Euler < 1/(3 * qn^2) && sln  Euler > 0)), pconv[k] = ((Euler  slp < 2/(3 * qp^2) && Euler  slp > 0)  (slp  Euler < 1/(3 * qp^2) && slp  Euler > 0))*pp + ((Euler  sln < 2/(3 * qn^2) && Euler  sln > 0)  (sln  Euler < 1/(3 * qn^2) && sln  Euler > 0))*pn; qconv[k] = ((Euler  slp < 2/(3 * qp^2) && Euler  slp > 0) 
(slp  Euler < 1/(3 * qp^2) && slp  Euler > 0))*qp + ((Euler  sln < 2/(3 * qn^2) && Euler  sln > 0)  (sln  Euler < 1/(3 * qn^2) && sln  Euler > 0))*qn; fract[k] = pconv[k]/qconv[k]; cf[k] = ((Euler  slp < 2/(3 * qp^2) && Euler  slp > 0)  (slp  Euler < 1/(3 * qp^2) && slp  Euler > 0))*cfm  ((Euler  sln < 2/(3 * qn^2) && Euler  sln > 0)  (sln  Euler < 1/(3 * qn^2) && sln  Euler > 0))*cfm; tst = 1, cfm = cfm + 1)); write("eulwritefile.txt", "Convergents: ", fract); write("eulwritefile.txt", "continued fraction: ", cf); write("eulwritefile.txt", "sln: ", sln); write("eulwritefile.txt", "slp: ", slp))


CROSSREFS

Cf. A002852.
Sequence in context: A094758 A283971 A100394 * A156671 A072006 A014238
Adjacent sequences: A178780 A178781 A178782 * A178784 A178785 A178786


KEYWORD

sign


AUTHOR

Joseph G. Johnson (jjohnson1253(AT)hotmail.com), Jun 12 2010


STATUS

approved



