Michael Joseph - cp's OEIS frontend

A265582 Number of (unlabeled) connected loopless multigraphs such that the sum of the numbers of vertices and edges is n.

1, 1, 0, 1, 1, 2, 3, 6, 10, 21, 41, 87, 187, 423, 971, 2324, 5668, 14224, 36506, 95880, 257081, 703616, 1962887, 5578529, 16137942, 47492141, 142093854, 432001458, 1333937382, 4181500703, 13301265585, 42918900353, 140423545125, 465712099790, 1565092655597

Offset: 0

Michael Joseph, Dec 10 2015

nonn

Also the number of connected skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below.
a(n) can be computed from A265580 and/or A265581, and partitions of n, by taking all loopless multigraphs (V,E) with |V| + |E| = n and subtracting out the disconnected ones.
a(n) <= A265580(n) except when n=1, and a(n) < A265580(n) for n>=6.

			For n = 5, the a(5) = 2 such multigraphs are the graph with three vertices and edges from one vertex to each of the other two, and the graph with two vertices connected by three edges.

Andrew Howroyd, Table of n, a(n) for n = 0..100
D. Einstein, M. Farber, E. Gunawan, M. Joseph, M. Macauley, J. Propp and S. Rubinstein-Salzedo, Noncrossing partitions, toggles, and homomesies, arXiv:1510.06362 [math.CO], 2015.

Cf. A191646, A265580, A265581.

\\ See A191646 for G, InvEulerMT.
seq(n)={my(v=InvEulerMT(vector((n+1)\2, k, 1 + y*Ser(G(k, n-1), y)))); Vec(1 + sum(i=1, #v, v[i]*y^i) + O(y*y^n))} \\ Andrew Howroyd, Feb 01 2020

From Andrew Howroyd, Feb 01 2020: (Start)
a(n) = Sum_{k=1..ceiling(n/2)} A191646(n-k, k) for n > 0.
Inverse Euler transform of A265581. (End)

Terms a(19) and beyond from Andrew Howroyd, Feb 01 2020

A265581 Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 56, 110, 222, 465, 1003, 2226, 5101, 12010, 29062, 72200, 183886, 479544, 1279228, 3486584, 9699975, 27520936, 79563707, 234204235, 701458966, 2136296638, 6611816700, 20784932424, 66333327604, 214819211047, 705650404444, 2350231740975

Offset: 0

Michael Joseph, Dec 10 2015

nonn

Also the number of skeletal 2-cliquish graphs with n vertices. See Einstein et al. link below.

a(n) is the sum of A265580(k) as k ranges from 0 to n. This is because there is a bijection between loopless multigraphs (V,E) satisfying |V| + |E| = k with no isolated vertices and loopless multigraphs (V,E) satisfying |V| + |E| = n with exactly n-k isolated vertices.

			For n = 4, the a(4) = 3 such multigraphs are the graph with four isolated vertices, the graph with three vertices and an edge between two of them, and the graph with two vertices connected by two edges.

Andrew Howroyd, Table of n, a(n) for n = 0..100
D. Einstein, M. Farber, E. Gunawan, M. Joseph, M. Macauley, J. Propp and S. Rubinstein-Salzedo, Noncrossing partitions, toggles, and homomesies, arXiv:1510.06362 [math.CO], 2015.

Cf. A191646, A192517, A265580, A265582.

\\ Needs G from A191646.
seq(n)={vector(n+1,i,1) + sum(k=1, n, concat(vector(n-k+1), G(n-k, k)))} \\ Andrew Howroyd, Feb 01 2020

a(n) = Sum_{k=0..n} A265580(k).
From Andrew Howroyd, Feb 01 2020: (Start)
a(n) = Sum_{i=1..n} A192517(i, n-i) for n > 0.
Euler transform of A265582. (End)

Terms a(19) and beyond from Andrew Howroyd, Feb 01 2020

A265580 Number of (unlabeled) loopless multigraphs with no isolated vertices such that the sum of the numbers of vertices and edges is n.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 4, 7, 13, 27, 54, 112, 243, 538, 1223, 2875, 6909, 17052, 43138, 111686, 295658, 799684, 2207356, 6213391, 17820961, 52042771, 154640528, 467254731, 1434837672, 4475520062, 14173115724, 45548395180, 148485883443, 490831193397, 1644581336531

Offset: 0

Michael Joseph, Dec 10 2015

nonn

Also the number of skeletal 2-cliquish graphs with n vertices and no isolated vertices. See Einstein et al. link below.

			For n = 5, the a(5) = 2 such multigraphs are the graph with three vertices and edges from one vertex to each of the other two, and the graph with two vertices connected by three edges.

Andrew Howroyd, Table of n, a(n) for n = 0..100
D. Einstein, M. Farber, E. Gunawan, M. Joseph, M. Macauley, J. Propp and S. Rubinstein-Salzedo, Noncrossing partitions, toggles, and homomesies, arXiv:1510.06362 [math.CO], 2015.

Cf. A265581.

a(n) = A265581(n) - A265581(n-1), n>=1.

Terms a(19) and beyond from Andrew Howroyd, Feb 01 2020

A126755 Braille numberdromes: numbers which read the same backwards and forwards in Braille.

Original entry on oeis.org

1, 2, 3, 7, 11, 22, 33, 46, 59, 64, 77, 80, 95, 111, 121, 131, 161, 171, 212, 222, 232, 262, 272, 313, 323, 333, 373, 416, 426, 436, 476, 519, 529, 539, 579, 614, 624, 634, 674, 717, 727, 737, 777, 810, 820, 830, 870, 915, 925, 935, 975

Offset: 1

Michael Joseph Halm, Apr 23 2007

The pairs 4 and 6, 5 and 9 and 0 and 8 are mirror images of each other. When a Braille number is read backward (as a mirror-image) the number is usually not the same as the original. Those that are the same could be called by analogy with the ordinary numberdromes the Braille numberdromes. Those with "a double yolk", such as 1081, would be Braille numberddromes, by analogy with palinddromes.

			a(8) = 46 because in Braille 4 and 6 are mirror images of each other.

Cf. A121018.

A101196 Position of n-th n after the decimal point in Pi.

Original entry on oeis.org

1, 16, 17, 36, 48, 72, 96, 74, 55, 854, 709, 1080, 1076, 1636, 1657, 1651, 889, 1674, 1227, 2039, 1486, 2372, 2690, 2288, 2033, 2282, 1785, 2703, 4155, 3102, 3584, 3767, 4325, 3808, 3551, 4081, 3785, 3229, 4464, 4884, 4127, 4228, 5336, 3961, 4242, 3633

Offset: 1

Michael Joseph Halm, Dec 12 2004

			a(2) = 16 because the second occurrence of 2 in the digits of pi after its decimal point is at position 16, that is, after 141592653589793.

Cf. A014777, A037000, A037001, A037002, A037003, A037004, A037005, A037006, A037007, A037008.

Corrected and extended by Mark Hudson (mrmarkhudson(AT)hotmail.com), Dec 13 2004

A102045 Aronsonian sequence generated by the sentence, "H is the first, fifth, sixteenth, twenty-fifth, thirty-sixth, thirty-eighth, forty-seventh, ... letter in this sentence.".

Original entry on oeis.org

1, 5, 16, 25, 36, 38, 47, 49, 57, 59, 71, 81, 93, 103, 119, 133, 140, 148, 155, 171, 183, 189, 196, 200, 204, 229, 231, 235, 251, 255, 277, 287, 291, 298, 308, 315, 319, 335, 339, 347, 351, 363, 367, 384, 388, 396, 409, 417, 426, 430, 450, 466, 470, 490, 494

Offset: 1

Michael Joseph Halm, Feb 12 2005

This version counts hyphens, A055508 does not.

			a(3) = 16 because the third H in "H is the first, fifth, ... letter in this sentence." is its 16th letter.

Cf. A005224.

A102046 Smallest positive integer greater than a(n - 1) consistent with the condition that n is a member of the sequence if and only if a(n) is congruent to (n!)!.

Original entry on oeis.org

1, 1, 2, 6, 7, 8, 720, 721, 722, 723, 724, 726, 727, 728, 729, 780, 781, 782, 783, 784, 785, 786, 787, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799

Offset: 0

Michael Joseph Halm, Feb 12 2005

The sequence is related to the fake even and fake odd sequences and also the factorial and double factorial sequences, so seems in the short run linear but in the long run exponential.

			a(6) = 720 because (3!)! = 6! = 720

Cf. A080591, A080588, A000197, A000142.

a(a(n)) = (n!)!

The definition does not match the data. How was this sequence generated? - N. J. A. Sloane, Feb 21 2021

A102069 Analogous to the oblong (promic or heteromecic) sequence formed but with reversal digits of factors multiplied.

Original entry on oeis.org

2, 6, 12, 20, 30, 42, 56, 72, 9, 11, 231, 651, 1271, 2091, 3111, 4331, 5751, 7371, 182, 24, 264, 704, 1344, 2184, 3224, 4464, 5904, 7544, 276, 39, 299, 759, 1419, 2279, 3339, 4599, 6059, 7719, 372, 56, 336, 816, 1496, 2376, 3456, 4736, 6216, 7896, 470, 75, 375

Offset: 1

Michael Joseph Halm, Feb 12 2005

Cf. A001129, A002378.

r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Table[r[n]*r[n+1],{n,51}] (* James C. McMahon, Jan 07 2024 *)

a(n) = R(n)*R(n+1), where R(n) is the digital reversal of n.

A098129 Concatenate 1 once, 2 twice, 3 three times, up to n n times.

Original entry on oeis.org

1, 122, 122333, 1223334444, 122333444455555, 122333444455555666666, 1223334444555556666667777777, 122333444455555666666777777788888888, 122333444455555666666777777788888888999999999

Offset: 1

Michael Joseph Halm, Jan 04 2005

a(n) is composite for all 2 <= n <= 1000. - David Cleaver, Mar 22 2023

			a(4) = 1223334444 because 1 concatenated once then concatenated with 2 twice and 3 three times and 4 four times gives 1223334444.

Seiichi Manyama, Table of n, a(n) for n = 1..31

Cf. A000461, A300517, A361751 (number of decimal digits).

a:= n-> parse(cat(seq(i$i, i=1..n))):
seq(a(n), n=1..12);  # Alois P. Heinz, Mar 07 2018

nn = 12; a[0] = {}; Do[Set[a[n], Join[a[n - 1], Flatten@ ConstantArray[IntegerDigits[n], n]]], {n, nn}]; Array[FromDigits @* a, nn] (* Michael De Vlieger, Mar 29 2023 *)

a(n) = {my(a=0,i,k);
  for(i=1,n, k = logint(i,10)+1;
    a = a*10^(i*k) + i*(10^(i*k)-1)/(10^k-1);
); return(a); } \\ David Cleaver, Mar 29 2023

def A098129(n): return int(''.join(str(j)*j for j in range(1,n+1))) # Chai Wah Wu, Mar 29 2023

Offset and a(8) corrected by Seiichi Manyama, Mar 07 2018

A095762 Numbers whose name in English contains an "h".

Original entry on oeis.org

3, 8, 13, 18, 23, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43, 48, 53, 58, 63, 68, 73, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 93, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Offset: 1

Michael Joseph Halm, Jul 10 2004

			a(1) = 3 because "three" contains an "h", while 0, 1 and 2 do not

cp's OEIS Frontend

User: Michael Joseph

A265582 Number of (unlabeled) connected loopless multigraphs such that the sum of the numbers of vertices and edges is n.

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A265581 Number of (unlabeled) loopless multigraphs such that the sum of the numbers of vertices and edges is n.

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A265580 Number of (unlabeled) loopless multigraphs with no isolated vertices such that the sum of the numbers of vertices and edges is n.

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A126755 Braille numberdromes: numbers which read the same backwards and forwards in Braille.

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A101196 Position of n-th n after the decimal point in Pi.

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A102045 Aronsonian sequence generated by the sentence, "H is the first, fifth, sixteenth, twenty-fifth, thirty-sixth, thirty-eighth, forty-seventh, ... letter in this sentence.".

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A102046 Smallest positive integer greater than a(n - 1) consistent with the condition that n is a member of the sequence if and only if a(n) is congruent to (n!)!.

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A102069 Analogous to the oblong (promic or heteromecic) sequence formed but with reversal digits of factors multiplied.

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Mathematica

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A098129 Concatenate 1 once, 2 twice, 3 three times, up to n n times.

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