A077467 -id:A077467 - cp's OEIS frontend

A077465 Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.

1, 3, 5, 11, 21, 43, 87, 173, 347, 693, 1387, 2775, 5549, 11099, 22197, 44395, 88789, 177579, 355159, 710317, 1420635, 2841269, 5682539, 11365079, 22730157, 45460315, 90920629, 181841259, 363682519, 727365037, 1454730075

Offset: 1

Eric W. Weisstein, Nov 05 2002

nonn

Harborth's recurrence can miss local minima that are 2 less than values in this sequence. A complete listing of cumulative minima is given by A084230.

H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A006046, A077464, A077466, A077467, A084230.

A077464 Stolarsky-Harborth constant; lim inf_{n->oo} F(n)/n^theta, where F(n) is the number of odd binomial coefficients in the first n rows and theta=log(3)/log(2).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 06 2002

The limit supremum of F(n)/n^theta is 1. - Charles R Greathouse IV, Oct 30 2016

Named by Finch (2003) after Kenneth B. Stolarsky and Heiko Harborth. Stolarsky (1977) evaluated that its value is in the interval [0.72, 0.815], and Harborth (1977) calculated the value 0.812556. - Amiram Eldar, Dec 03 2020

			0.812556559016006387694882...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 145-151.

Heiko Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc., Vol. 62, No. 1 (1977), pp. 19-22.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024. See pp. 2, 4.
Kenneth B. Stolarsky, Digital sums and binomial coefficients, Notices of the American Mathematical Society, Vol. 22, No. 6 (1975), A-669, entire volume.
Kenneth B. Stolarsky, Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity, SIAM J. Appl. Math., Vol. 32, No. 4 (1977), pp. 717-730.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant.
Eric Weisstein's World of Mathematics, Pascal's Triangle.

Cf. A006046, A020857, A077465, A077466, A077467.

Equals lim inf_{n->oo} A006046(n)/n^A020857. - Amiram Eldar, Dec 03 2020

A077466 a(n) = A006046(A077465(n)).

Original entry on oeis.org

1, 5, 11, 37, 103, 317, 967, 2869, 8639, 25853, 77623, 232997, 698735, 2096461, 6288871, 18867125, 56600351, 169802077, 509408279, 1528220741, 4584666319, 13753990765, 41261980487, 123785957845, 371357840767, 1114073555069

Offset: 1

Eric W. Weisstein, Nov 05 2002

nonn

H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A006046, A077464, A077465, A077467.

A195119 a(n) = 2n - floor(nsqrt(2)).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42

Offset: 0

Clark Kimberling, Sep 09 2011

Not A077467, which has only one 17; not A098294, which has two 41's.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A005843 (2n), A001951 (floor(n*sqrt(2))).

[2*n-Floor(n*Sqrt(2)): n in [0..70]]; // Vincenzo Librandi, Sep 12 2011

seq(2*n-floor(n*sqrt(2)),n=0..80); # Muniru A Asiru, Sep 30 2018

Table[2*n - Floor[n*Sqrt[2]], {n,0,100}] (* G. C. Greubel, Sep 29 2018 *)

vector(100, n, n--; 2*n-floor(n*sqrt(2))) \\ G. C. Greubel, Sep 29 2018

a(n) = A005843(n) - A001951(n). - Michel Marcus, Sep 30 2018

cp's OEIS Frontend

A077465 Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.

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A077464 Stolarsky-Harborth constant; lim inf_{n->oo} F(n)/n^theta, where F(n) is the number of odd binomial coefficients in the first n rows and theta=log(3)/log(2).

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Formula

A077466 a(n) = A006046(A077465(n)).

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A195119 a(n) = 2n - floor(nsqrt(2)).

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cp's OEIS Frontend

A077465 Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.

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A077464 Stolarsky-Harborth constant; lim inf_{n->oo} F(n)/n^theta, where F(n) is the number of odd binomial coefficients in the first n rows and theta=log(3)/log(2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A077466 a(n) = A006046(A077465(n)).

Views

Author

Keywords

Links

Crossrefs

A195119 a(n) = 2*n - floor(n*sqrt(2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A195119 a(n) = 2n - floor(nsqrt(2)).