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 A178783 Continued fraction for Euler-Mascheroni 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^2-1, 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 (list; graph; refs; listen; history; text; internal format)
 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 Euler-Maclaurin summation and gamma itself for the upper and lower bounds. LINKS PROG (PARI) pconv=vector(43); qconv=vector(43); cf=vector(43); fract=vector(43); pconv=0; pconv=1; pconv=1; pconv=4; qconv=1; qconv=1; qconv=2; qconv=7; cf=0; cf=1; cf=1; cf=3; fract=0/1; fract=1/1; fract=1/2; fract=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

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Last modified April 5 20:30 EDT 2020. Contains 333260 sequences. (Running on oeis4.)