A228639 -id:A228639 - cp's OEIS frontend

A123737 Partial sums of (-1)^floor(n*sqrt(2)).

-1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -1, -2, -3, -2, -1, -2, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0

Offset: 1

T. D. Noe, Oct 11 2006

Conjecture: A001652(n) is the index of the first occurrence of n in sequence A123737, A001108(n) is the index of the first occurrence of -n in sequence A123737. - Vaclav Kotesovec, Jun 02 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
Henk Bruin and Robbert Fokkink, On the records and zeros of a deterministic random walk, arXiv:2503.11734 [math.DS], 2025. See p. 2.
Kevin O'Bryant, Bruce Reznick and Monika Serbinowska, Almost alternating sums, arXiv:math/0308087 [math.NT], 2003-2205; Amer. Math. Monthly, Vol. 113 (October 2006), pp. 673-688.

Cf. A123724 (sum for 2^(1/3)), A123738 (sum for Pi), A123739 (sum for e).

Cf. A001652, A001951, A228639.

[&+[(-1)^Floor(j*Sqrt(2)): j in [1..n]]: n in [1..130]]; // G. C. Greubel, Sep 05 2019

ListTools:-PartialSums([seq((-1)^floor(n*sqrt(2)),n=1..100)]); # Robert Israel, Jun 02 2015

Rest[FoldList[Plus,0,(-1)^Floor[Sqrt[2]*Range[120]]]]
Accumulate[(-1)^Floor[Range[120]Sqrt[2]]] (* Harvey P. Dale, Jan 16 2012 *)
(* The positions of the first occurrences of n and -n in this sequence: *) stab = Rest[FoldList[Plus,0,(-1)^Floor[Sqrt[2]*Range[1000000]]]]; Print[Table[FirstPosition[stab,n][[1]],{n,1,8}]]; Print[Table[FirstPosition[stab,-n][[1]],{n,1,8}]]; (* Vaclav Kotesovec, Jun 02 2015 *)

a(n)=sum(i=1,n,(-1)^sqrtint(2*i^2)) \\ Charles R Greathouse IV, Feb 07 2013

[sum((-1)^floor(j*sqrt(2)) for j in (1..n)) for n in (1..130)] # G. C. Greubel, Sep 05 2019

O'Bryant, Reznick, & Serbinowska show that |a(n)| <= k log n + 1, with k = 1/(2 log (1 + sqrt(2))), and further -a(n) > k log n + 0.78 infinitely often. - Charles R Greathouse IV, Feb 07 2013

A339799 Decimal expansion of Sum_{m>=1} (-1)^floor(sqrt(m)) / m.

Original entry on oeis.org

1, 2, 9, 4, 0, 8, 1, 2, 2, 1, 8, 8, 3, 0, 9, 1, 0, 7, 6, 3, 0, 3, 8, 2, 1, 7, 1, 8, 3, 5, 6, 7, 3, 1, 2, 5, 0, 5, 0, 1, 1, 2, 2, 5, 9, 5, 3, 9, 9, 2, 0, 4, 3, 0, 2, 2, 7, 6, 5, 9, 2, 3, 3, 9, 5, 2, 7, 5, 5, 1, 7, 1, 2, 7, 9, 3, 8, 5, 1, 5, 7, 1, 2, 0, 9, 0, 3, 6, 2, 6, 1, 8, 4, 8, 6, 1, 4, 2, 7, 8, 9, 6, 0, 8, 2

Offset: 1

Bernard Schott, Dec 17 2020

When grouped by negative and positive packs = - (1+1/2+1/3) + (1/4+1/5+1/6+1/7+1/8) - (1/9+...+1/15) + (1/16+...+1/24) +...+ (-1)^k (1/k^2 +...+ 1/((k+1)^2-1)) + ...

Sum_{m>=1} (-1)^floor(sqrt(m)) / m^q is convergent iff q > 1/2.

			-1.2940812218830910763038217183567312505011225953992043022765923395275517127938...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.35, p. 287.
E. Ramis , C. Deschamps, J. Odoux, Analyse 2, Exercices avec solutions, Classes Préparatoires aux Grandes Ecoles Scientifiques, Masson, Paris, 1985, Exercice 1. 1.14, pp. 12-13.

Wikipedia, Digamma function.

Cf. A196521, A228639, A231902.

evalf(Sum((-1)^n*(Psi(n^2 + 2*n + 1) - Psi(n^2)), n = 1 .. infinity), 120); # Vaclav Kotesovec, Dec 18 2020

sumalt(k=1, (-1)^k * (psi(1 + 2*k + k^2) - psi(k^2))) \\ Vaclav Kotesovec, Dec 18 2020

Equals Sum_{m>=1} (-1)^floor(sqrt(m)) / m.
Equals Sum_{m>=1} (-1)^m * Sum_{k=m^2..(m+1)^2-1} 1/k.
Equals Sum_{m>=1} (-1)^m * (digamma((m+1)^2) - digamma(m^2)).

More terms from Vaclav Kotesovec, Dec 18 2020

A289252 Decimal expansion of the mean number of iterations in a comparison algorithm using centered continued fractions, a constant related to Vallée's constant.

Original entry on oeis.org

1, 0, 8, 9, 2, 2, 1, 4, 7, 3, 8, 6

Offset: 1

Jean-François Alcover, Sep 02 2017

From Jon E. Schoenfield, Jan 27 2018: (Start)
If we define the partial sum s_k = (360/Pi^4) * Sum_{i..k} Sum_{j=ceiling(phi*i)..floor((phi+1)*i)} 1/(i^2*j^2), then the real-valued sequence s_0, s_1, s_2, s_3, ... converges very slowly, and the convergence is not smooth because of the aperiodicity created by the presence of the functions ceiling(phi*i) and floor((phi+1)*i) in the limits on j in the inner sum. However, if we define the partial sum S_k = s_Fibonacci(k), then the real-valued sequence S_0, S_1, S_2, S_3, ... converges fairly quickly. (Cf. A228639.)
Also, the subsequences S_Even = {s_0, s_1, s_3, s_8, s_21, ..., s_Fibonacci(2*d), ...} for d >= 0 and S_Odd = {s_1, s_2, s_5, s_13, s_34, ..., s_Fibonacci(2*d+1), ...} for d >= 0 both converge to lim_{k->infinity} s_k = 1.08922147... in a way that can be accelerated using successive applications of Richardson extrapolation, and--given the values of s_Fibonacci(m) for m=0..27--appears to yield the limit 1.08922147386406851032218345320... (This would seem to indicate that the last two terms currently in the Data section are incorrect.) (End)

			1.08922147386...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's constant, p. 162.

Eric Weisstein's World of Mathematics, Vallée Constant.

Cf. A074903, A143302.

terms = 10^6;
f[i_Integer] := f[i] = NSum[1/(i^2*j^2), {j, Ceiling[ GoldenRatio * i], Floor[(1 + GoldenRatio) * i]}, WorkingPrecision -> 30];
s = 360/Pi^4 * NSum[f[i], {i, 1, Infinity}, Method -> "WynnEpsilon", NSumTerms -> terms];
RealDigits[s, 10, 12][[1]] (* updated Jun 14 2019 *)

Equals (360/Pi^4) * Sum_{i >= 1} Sum_{j=ceiling(phi*i)..floor((phi+1)*i)} 1/(i^2*j^2).

Corrected and extended to 12 digits by Jean-François Alcover, Jun 14 2019, after Jon E. Schoenfield's pertinent comment.

A351738 Decimal expansion of Sum_{k>0} sin(sqrt(k)) / k.

Original entry on oeis.org

1, 7, 1, 5, 6, 7, 1, 7, 9, 4, 7, 0, 9

Offset: 1

Bernard Schott, May 20 2022

Sum_{k>0} sin(k^alpha) / (k^beta) with 0 < alpha < 1 is convergent if beta > max(alpha, 1-alpha); the constant of this sequence corresponds to the case alpha = 1/2 and beta = 1 (see Arnaudiès).
Consequence: Sum_{k>0} sin(k^(1/m)) / k converges for any positive integer m.
The sequence converges slowly.

			1.715671794709...

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, 1993, Exercice 11, pp. 316-319.

Mathematics Stack Exchange, Convergence of Sum_{k=0..infinity} sin(sqrt(k)) / k.

Cf. A096418, A096444, A114940, A228639, A263193, A342680.

default(realprecision, 100); sumalt(k=0, sum(j=1+floor(k^2*Pi^2),floor((k+1)^2*Pi^2), sin(sqrt(j))/j)) \\ Vaclav Kotesovec, May 21 2022

More digits from Stefano Spezia, May 21 2022

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A123737 Partial sums of (-1)^floor(n*sqrt(2)).

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A339799 Decimal expansion of Sum_{m>=1} (-1)^floor(sqrt(m)) / m.

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A289252 Decimal expansion of the mean number of iterations in a comparison algorithm using centered continued fractions, a constant related to Vallée's constant.

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Mathematica

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A351738 Decimal expansion of Sum_{k>0} sin(sqrt(k)) / k.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Extensions