A095270 -id:A095270 - cp's OEIS frontend

A095110 Sum of maximal Motzkin path prefix-lengths of all 4k+3 integers in range ]2^n,2^(n+1)].

2, 6, 24, 80, 303, 847, 3401, 12816, 47255, 168306, 628058, 2243714, 8741445, 33079519, 125266470, 474015314, 1806320048, 6945673364, 26539784794, 102630867522

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095270, A095109.

A095269 Diving index of 4n+3.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 10, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 3, 0, 7, 7, 0, 0, 3, 0, 8, 0, 8, 0, 3, 0, 0, 0, 11, 0, 3, 0, 34, 0, 8, 19, 3, 0, 10, 13, 61, 0, 3, 0, 0, 7, 0, 0, 3, 0, 7, 0, 8, 0, 3, 0, 10, 0, 0, 0, 3, 0, 32, 0, 7, 0, 3, 0, 58, 0, 7, 0, 3, 0, 0

Offset: 0

Antti Karttunen, Jun 01 2004

nonn

Diving index of an odd number n is the first integer u > 1 where Sum_{i=1..u} J(i/n) results -1 and zero if never. Here J(i/n) is Jacobi symbol of i and n, which reduces to a Legendre symbol L(i/n) when n is a prime.

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A004767, A095270, A095109, A095271 (same sequence with zeros removed).

a(n) = A095270(n)+1 modulo A004767(n).

A095105 Length of maximal Dyck path prefix in the Legendre-vector of the n-th 4k+3 prime (A002145(n)).

Original entry on oeis.org

2, 6, 10, 2, 22, 30, 2, 46, 58, 2, 70, 78, 82, 102, 6, 6, 130, 2, 150, 2, 166, 10, 190, 198, 2, 12, 60, 238, 250, 262, 270, 2, 2, 310, 2, 6, 358, 44, 2, 382, 418, 430, 438, 6, 6, 34, 478, 6, 34, 2, 502, 2, 2, 562, 2, 14, 598, 606, 2, 14, 2, 646, 658, 6, 2, 718

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

a(n)=A095104(n)-1 modulo A002145(n). Cf. A095107, A095270.

A095361 Ratio A095110(n)/A000079(n-2) rounded down.

Original entry on oeis.org

2, 6, 12, 20, 37, 52, 106, 200, 369, 657, 1226, 2191, 4268, 8076, 15291, 28931, 55124, 105982, 202482, 391505

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

This is the average length of maximum Motzkin path prefix (i.e. non-diving portion) found in the "Jacobi-vectors" of all 4k+3 integers in range ]2^n,2^(n+1)]. See A095269-A095270.

The ratios before rounding are: 2, 6, 12, 20, 37.875, 52.9375, 106.28125, 200.25, 369.179687, 657.445312, 1226.675781, 2191.126953, 4268.283691, 8076.054443, 15291.317139, 28931.598755, 55124.513184, 105982.564758, 202482.488968, 391505.689705.

A095362 gives the same ratios rounded to nearest integer. A095357 gives similar ratios computed only for 4k+3 primes.

a(1) = 2, a(n) = floor(A095110(n)/A000079(n-2)) for n > 1.

cp's OEIS Frontend

A095110 Sum of maximal Motzkin path prefix-lengths of all 4k+3 integers in range ]2^n,2^(n+1)].

Views

Author

Keywords

Links

Crossrefs

A095269 Diving index of 4n+3.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A095105 Length of maximal Dyck path prefix in the Legendre-vector of the n-th 4k+3 prime (A002145(n)).

Views

Author

Keywords

Links

Crossrefs

A095361 Ratio A095110(n)/A000079(n-2) rounded down.

Views

Author

Keywords

Comments

Crossrefs

Formula