Joseph Wheat - cp's OEIS frontend

A330285 The maximum number of arithmetic progressions in a sequence of length n.

0, 0, 1, 3, 7, 12, 20, 29, 41, 55, 72, 90, 113, 137, 164, 194, 228, 263, 303, 344, 390, 439, 491, 544, 604, 666, 731, 799, 872, 946, 1027, 1109, 1196, 1286, 1379, 1475, 1579, 1684, 1792, 1903, 2021, 2140, 2266, 2393, 2525, 2662, 2802, 2943, 3093, 3245, 3402, 3562, 3727

Offset: 1

Joseph Wheat, Dec 21 2019

nonn

The partial arithmetic density D_n(A) up to n is merely the number of arithmetic progressions, A(s(n)), divided by the total number of nonempty subsets of {s(1), s(2), ..., s(n)}, i.e., A(s(n))/(2^n - 1). As n approaches infinity, D_n(A) converges to zero. Furthermore, the infinite sum of the partial densities for any sequence always converges to the total density D(A). Every infinite arithmetic progression has the same total density, Sum_{n >= 1} a(n)/(2^n - 1) = alpha ~ 1.25568880818612911696845537; sequences with a finite number of progressions have D(A) < alpha; and sequences without any arithmetic progressions have D(A) = 0.

Encyclopedia of Mathematics, Density of a sequence
Eric Weisstein's World of Mathematics, Arithmetic Progression

Cf. A049988, A130518, A104429, A065825, A092482.

Partial sums of A002541.

a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).

A316261 The number of ways to induce a single pinch on a compact 2-manifold with n handles. (Note: The manifold is embedded in Euclidean 2-space, and each pinch partitions it into at most two submanifolds.)

Original entry on oeis.org

1, 3, 9, 15, 26, 37, 55, 73, 100, 127, 165, 203, 254, 305, 371, 437, 520, 603, 705, 807, 930, 1053, 1199, 1345, 1516, 1687, 1885, 2083, 2310, 2537, 2795, 3053, 3344, 3635, 3961, 4287, 4650, 5013, 5415, 5817, 6260

Offset: 0

Joseph Wheat, Jun 27 2018

The formula for this sequence can be derived by separating the conformed manifolds into three sets. The first set consists of those conformations where the handles of the manifold are pinched at the boundary, the second set have two or more handles pinched at the interior of the manifold, and the third set are pinched at the boundary and may or may not have handles drawn into this pinch. The order of the first set is n, the order of the second is n - 1, and the order of the third set is given by the following series: (Sum_{k mod 2 = 0..n} (k/2)*(n - k + 1) + (2*(n - k) + (-1)^(n - k) + 3)/4) + (Sum_{j mod 2 = 1..n} ((j + 1)/2)*(n - j + 1)). These can then be combined into a single expression, Sum_{i = 0..n} ((2*i + (-1)^(i + 1) + 1)/4)*(n - i + 1) + ((2*(n - i) + (-1)^(n - i) + 3)/4)*(((-1)^i + 1)/2). The i in this series can be thought of as the number of handles drawn into the central pinch. If one factors out the expressions in the series and simplifies each term individually, the resulting functions can then be combined into a single formula. However, when we add 2n - 1 to this we find that for n = 0 the formula also equals zero. This cannot be, because there is one way to pinch a compact 2-manifold with 0 handles. Therefore, ((-1)^(2^n - 1) + 1)/2 is added as a corrective term for this one case.

			For a visual example see links.

Jonathan L. Gross, Jay Yellen, and Ping Zhang, The Handbook of Graph Theory (Second Edition), CRC Press, 2013, pp. 730-806.
Ana Claudia Nabarro, Juan J. Nuño-Ballesteros, Raúl Oset Sinha, Maria Aparecida Soares Ruas, Contemporary Mathematics: Real and Complex Singularities, American Mathematical Soc., 2014, pp. 50-51.

Colin Barker, Table of n, a(n) for n = 0..1000
Jonathan L. Gross, Jay Yellen, and Ping Zhang, The Handbook of Graph Theory (Second Edition)
Allen Hatcher, Algebraic Topology (Ch. 0)
Ana Claudia Nabarro, Juan J. Nuño-Ballesteros, Raúl Oset Sinha, Maria Aparecida Soares Ruas, Contemporary Mathematics: Real and Complex Singularities
Pseudomanifold. The Encyclopedia of Mathematics
Joseph Wheat, Visual Example of n = 3
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A087811.

a[n_] := (2 n^3 + 12 n^2 + 73 n + 3 (n + 2)*(-1)^n - 6)/24 + ((-1)^(2^n - 1) + 1)/2; Array[a, 50, 0] (* or *)
CoefficientList[ Series[(x^6 + x^5 - 2x^4 - 2x^3 + 2x^2 + x + 1)/((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Robert G. Wilson v, Jul 23 2018 *)

Vec((1 + x + 2*x^2 - 2*x^3 - 2*x^4 + x^5 + x^6) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jul 05 2018

a(n) = (2*n^3 + 12*n^2 + 73*n + 3*(n + 2)*(-1)^n - 6)/24 + ((-1)^(2^n - 1) + 1)/2.
From Colin Barker, Jul 05 2018: (Start)
G.f.: (1 + x + 2*x^2 - 2*x^3 - 2*x^4 + x^5 + x^6) / ((1 - x)^4*(1 + x)^2).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.
(End)

A299404 a(n) = 1 + Sum_{m >= 1} (m + 1)^n/2^(m - 1).

Original entry on oeis.org

3, 7, 23, 103, 599, 4327, 37463, 378343, 4366679, 56698087, 817980503, 12981060583, 224732540759, 4214866787047, 85130743763543, 1842265527822823, 42525237455850839, 1042966136233087207, 27084277306054762583, 742412698554627289063, 21421502369955073624919

Offset: 0

Joseph Wheat, Feb 20 2018

nonn

Cf. A007047, A083652, A162509.

Table[1 + LerchPhi[1/2, -n, 2], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2018 *)

a(n) = 1+ round(suminf(m=1, (m + 1)^n/2^(m - 1)));

a(n + 1) = 4*A162509(n + 1) + a(n).
a(n) = 2*A007047(n) + 1.
{a(4n - 3), a(4n - 2), a(4n - 1), a(4n)} mod 10 = {7, 3, 3, 9} for n > 0.
floor(log_2(a(n))) = A083652(n).
Lim_{n->infinity} (a(n)^(1/n))/n = 1/(e*log(2)). - Jon E. Schoenfield, Feb 24 2018
a(n)/n! ~ 4 / (log(2))^(n+1). - Vaclav Kotesovec, Apr 17 2018

A296515 Number of edges in a maximal planar graph with n vertices.

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 0

Joseph Wheat, Jonathan Sondow, Feb 28 2018

{a(n)} is a monotonic increasing sequence because a maximal planar graph of order n can be generated on n + 1 nodes. Therefore a(n) <= a(n + 1).
Maximal planar graphs of order n > 5 are not unique.
|E(G_2n)| = (2n - 1) + 2*Sum_{k=0..(floor(log_2(n - 1)))} floor((n - 1)/2^k) where |E(G_2n)| is the size of a minimal planar graph G of order 2n.
Number of edges of a maximal 3-degenerate graph of order n (this class includes 3-trees).  The intersection of this class and maximal planar graphs is the Apollonian networks (planar 3-trees); neither class contains the other. - Allan Bickle, Nov 14 2021
a(n) is the number of blocks to dig (in a staircase fashion) to get out of a hole of depth n in Minecraft. - Max R Anderson, Oct 19 2023

Stefano Spezia, Table of n, a(n) for n = 0..10000
Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Dawei He, Yan Wang, and Xingxing Yu, The Kelmans-Seymour conjecture I, arXiv:1511.05020 [math.CO], 2015.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
Mathematics Stack Exchange, You dig a hole in Minecraft straight down n blocks, how many blocks must you dig to get out?
K. Stephenson, Circle Packing: A Mathematical Tale, Notices Amer. Math. Soc. 50 (2003).
Squid Tamar-Mattis, Planar Graphs and Wagner's and Kuratowski's Theorems, REU (2016).
Eric Weisstein's World of Mathematics, Kuratowski Reduction Theorem, Polyhedral Graph
David R. Wood, A Simple Proof of the Fary-Wagner Theorem, arXiv:cs/0505047 [math.CO], 2005.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008486, A008585, A242088.

CoefficientList[Series[(x^2 (1 + x + x^2))/(x - 1)^2, {x, 0, 60}], x] (* or *) LinearRecurrence[{2, -1}, {0, 0, 1, 3}, 60] (* Robert G. Wilson v, Mar 04 2018 *)

a(n) = floor(6/2^n) + 3n - 6 (see comments section of A008486).
G.f.: x^2 + 3*x^3/(x - 1)^2. - R. J. Mathar, Apr 14 2018
E.g.f.: 6 + x*(x + 6)/2 + 3*exp(x)*(x - 2). - Stefano Spezia, Feb 13 2023
a(n) = 3*(n - 2) for n >= 3. - Max R Anderson, Oct 19 2023

A290474 Number of fractional partitions of n where each element of a partition is a rational number r/s such that r < s, s <= n, and gcd(r,s) = 1.

Original entry on oeis.org

1, 0, 1, 12, 135, 4477, 100160, 8663934, 485380025, 80730951180, 10180011676356, 4126137351376215, 563950787766342780, 456369006693283278869, 200330760220853808357439, 335435016971402890883460861, 197675615401466868237710861644, 561969529551274362018496511765678

Offset: 0

Joseph Wheat, Aug 03 2017

nonn

a(n) = (n^2 + 1)^(-1 + Sum_{k=1..n} phi(k)) - f(n) where phi(n) is Euler's totient function, and f(n) is the number of trivial solutions which do not satisfy the equation q_1*x_1 + q_2*x_2 + ... + q_m*x_m = n. Each coefficient is a rational number satisfying the criteria given in the definition, and m = -1 + Sum_{k=1..n} phi(k).

			For n=3, the number of partitions is equal to the number of nonnegative integer solutions for the equation: (1/2)*x_1 + (1/3)*x_2 + (2/3)*x_3 = 3. The set S of solutions is {[0,1,4], [0,3,3], [0,5,2], [0,7,1], [0,9,0], [2,0,3], [2,2,2], [2,4,1], [2,6,0], [4,1,1], [4,3,0], [6,0,0]}. Therefore, |S| = a(3) = 12.

Cf. A015614, A057562, A154886, A154887, A171978.

s(v, n, t) = {if(t==#v, f = n\v[t]; v[t]*f == n, sum(i=0, n\v[t], s(v, n-v[t]*i, t+1)))}
a(n) = {if(n<=2, return(n-1)); my(fractions = List(), q = 0); for(i=2, n, for(j=1, i-1, if(gcd(i, j)==1, listput(fractions, j/i)))); s(fractions, n, 1)} \\ David A. Corneth, Aug 03 2017

a(7)-a(17) from Alois P. Heinz, Aug 03 2017

A290025 The partial sums of 2^d(n) where d(n) is the n-th digit of the concatenated triangular numbers, and d(1)=0.

Original entry on oeis.org

1, 3, 11, 75, 77, 78, 80, 112, 116, 118, 122, 378, 386, 450, 466, 498, 530, 562, 626, 690, 818, 1074, 1586, 1588, 1590, 1591, 1623, 1625, 1629, 1630, 1632, 1640, 1704, 1706, 1738, 1746, 1748, 1876, 1878, 1880, 2392, 2393, 2397, 2399, 2400, 2404, 2412, 2414, 2418, 2450

Offset: 1

Joseph Wheat, Jul 17 2017

The differences between consecutive terms are <= 2^9. So the sequence contains arbitrarily long arithmetic progressions. The sequence of powers of 2 does not contain progressions, however. This is a result of the fact that 2^n satisfies the recurrence relation a(n+1)=2a(n).

			2^d(1) + 2^d(2) + 2^d(3) = 2^0 + 2^1 + 2^3 = 11.

Cf. A000217, A034004.

Accumulate[2^Flatten@ Map[IntegerDigits, Array[# (# + 1)/2 &, 23, 0]]] (* Michael De Vlieger, Aug 03 2017 *)

lista(nn) = {print1(cur=1, ", "); for(n=1, nn, d = digits(n*(n+1)/2); for(i=1, #d, cur += 2^d[i]; print1(cur, ", ");););} \\ Michel Marcus, Jul 21 2017

first(n) = {my(d = [0], i = 1, t = 2, res = vector(n)); res[1] = 1; while(#d < n, d = concat(d, digits(i)); i+=t; t++); for(i=2, n, res[i] = res[i-1] + 2^d[i]); res} \\ David A. Corneth, Aug 03 2017

a(n) = Sum_{k=1..n} 2^d(k) where d(k) = A034004(k).

More terms from Michel Marcus, Jul 21 2017

A289898 a(n) = floor((2^prime(n+1))/Sum_{k=0|n,2^prime(k)}).

Original entry on oeis.org

2, 2, 2, 11, 3, 12, 3, 12, 59, 3, 51, 15, 3, 12, 59, 62, 3, 51, 15, 3, 50, 15, 60, 251, 15, 3, 12, 3, 12, 15179, 15, 60, 3, 816, 3, 51, 62, 15, 60, 62, 3, 816, 3, 12, 3, 3226, 4094, 15, 3, 12, 59, 3, 816, 63, 63, 63, 3, 51, 15, 3, 808, 16363, 15, 3, 12, 15183

Offset: 1

Joseph Wheat, Jul 14 2017

nonn

Cf. A034785, A076793.

Table[Floor[2^Prime[n + 1]/Sum[ 2^Prime[k], {k, n}]], {n, 66}] (* Michael De Vlieger, Jul 16 2017 *)

a(n) = 2^prime(n+1)\sum(k=1, n, 2^prime(k)); \\ Michel Marcus, Jul 16 2017

a(n) = floor(2^prime(n+1)/(Sum_{k=1..n} 2^prime(k))).

A289836 Numbers k such that floor(e*k + Pi) is a square.

Original entry on oeis.org

5, 17, 29, 36, 52, 71, 132, 146, 177, 211, 229, 330, 648, 744, 956, 1112, 1368, 1413, 1459, 1506, 1700, 1906, 2124, 2295, 2657, 2720, 2848, 2913, 2979, 3181, 3319, 3532, 3678, 3902, 3978, 4055, 4211, 4290, 4780, 5035

Offset: 1

Joseph Wheat, Jul 13 2017

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Select[Range@ 5040, IntegerQ@ Sqrt@ Floor[# E + Pi] &] (* Michael De Vlieger, Jul 14 2017 *)

is(n)=issquare(floor(exp(1)*n+Pi)) \\ Charles R Greathouse IV, Jul 14 2017

from gmpy2 import is_square
from mpmath import *
mp.dps=100
def ok(n): return is_square(int(nint(floor(exp(1)*n + pi))))
print([n for n in range(1, 6001) if ok(n)]) # Indranil Ghosh, Jul 15 2017

More terms from Charles R Greathouse IV, Jul 14 2017

A289830 a(n) satisfies the equation n/(n-1) + a(n)/n! = H(n), where H(n) is the n-th harmonic number.

Original entry on oeis.org

-1, 2, 18, 124, 900, 7188, 63504, 618336, 6596640, 76635360, 963895680, 13056819840, 189581333760, 2938083321600, 48416639846400, 845487698227200, 15598004134809600, 303161985274982400, 6191998554470400000, 132599321499875328000, 2970952207377960960000

Offset: 2

Joseph Wheat, Jul 12 2017

sign

Cf. A001008, A002805.

Table[n!*(HarmonicNumber[n] - n/(n - 1)), {n, 2, 22}] (* Michael De Vlieger, Jul 13 2017 *)

from sympy import factorial, harmonic
def a(n): return factorial(n-2)*(harmonic(n)*(n-1) - n)*n
print([a(n) for n in range(2, 26)]) # Indranil Ghosh, Jul 14 2017

a(n) = n! * (H(n) - n/(n-1)). - Alois P. Heinz, Jul 13 2017

More terms from Alois P. Heinz, Jul 13 2017

A289829 Perfect squares of the form prime(k+1)^2 - prime(k)^2 + 1 where prime(k) is the k-th prime number.

Original entry on oeis.org

25, 49, 121, 169, 289, 361, 841, 961, 1681, 1849, 2401, 2809, 3721, 5929, 6889, 7921, 8281, 10201, 11449, 11881, 14161, 14641, 17689, 24649, 26569, 32041, 38809, 41209, 43681, 44521, 61009, 63001, 69169, 76729, 80089, 85849, 89401, 94249, 96721, 97969, 108241

Offset: 1

Joseph Wheat, Jul 12 2017

nonn

			7^2 - 5^2 + 1 = 5^2, 17^2 - 13^2 + 1 = 11^2, 47^2 - 43^2 + 1 = 19^2, etc.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

TakeWhile[#, # < 110000 &] &@ Union@ Select[Array[Prime[# + 1]^2 - Prime[#]^2 + 1 &, 10^4], IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Jul 13 2017 *)
Take[Select[#[[2]]-#[[1]]+1&/@Partition[Prime[Range[3000]]^2,2,1],IntegerQ[Sqrt[#]]&]//Union,50] (* Harvey P. Dale, Jan 19 2025 *)

is(n) = if(!issquare(n), return(0), my(p=2); while(1, if(n==nextprime(p+1)^2-p^2+1, return(1)); p=nextprime(p+1); if(p > n, return(0)))) \\ Felix Fröhlich, Jul 15 2017

from _future_ import division
from sympy import divisors, isprime, prevprime, nextprime
A289829_list = []
for n in range(10**4):
    m = n**2-1
    for d in divisors(m):
        if d*d >= m:
            break
        r = m//d
        if not r % 2:
            r = r//2
            if not isprime(r):
                p, q = prevprime(r), nextprime(r)
                if m == (q-p)*(q+p):
                    A289829_list.append(n**2)
                    break # Chai Wah Wu, Jul 15 2017

More terms from Alois P. Heinz, Jul 13 2017

cp's OEIS Frontend

User: Joseph Wheat

A330285 The maximum number of arithmetic progressions in a sequence of length n.

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A316261 The number of ways to induce a single pinch on a compact 2-manifold with n handles. (Note: The manifold is embedded in Euclidean 2-space, and each pinch partitions it into at most two submanifolds.)

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A299404 a(n) = 1 + Sum_{m >= 1} (m + 1)^n/2^(m - 1).

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A296515 Number of edges in a maximal planar graph with n vertices.

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A290474 Number of fractional partitions of n where each element of a partition is a rational number r/s such that r < s, s <= n, and gcd(r,s) = 1.

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A290025 The partial sums of 2^d(n) where d(n) is the n-th digit of the concatenated triangular numbers, and d(1)=0.

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A289898 a(n) = floor((2^prime(n+1))/Sum_{k=0|n,2^prime(k)}).

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A289836 Numbers k such that floor(e*k + Pi) is a square.

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