A002083 -id:A002083 - cp's OEIS frontend

A058222 Tree of tournament sequences read across rows.

1, 2, 3, 4, 4, 5, 6, 5, 6, 7, 8, 5, 6, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 7, 8, 9

Offset: 0

N. J. A. Sloane, Dec 02 2000

			Irregular triangle begins:
  1;
  2;
  3,4;
  4,5,6,5,6,7,8;
  ...

Sean A. Irvine, Table of n, a(n) for n = 0..10000
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

A008934 gives number of children at level n. Cf. A058223.

Cf. A002083.

Top node is 1; each node k has children labeled k+1, k+2, ..., 2k at next level.

A062178 a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936, 2657534, 5314400, 10628132, 21254942, 42508562, 85014494, 170026358, 340047481, 680089727, 1360169009, 2720327573

Offset: 0

Henry Bottomley, Jun 12 2001

nonn

As partial sum of Narayana-Zidek-Capell numbers A002083, this is the number of words beginning with 1, with sum of integers <=n, in the sequence 1, 11, 111, 112, 1111, 1112, 1113, 1121, 1122, 1123, 1124, 11111, 11112, 11113, 11114, 11121, 11122, 11123, 11124, 11125, 11131, 11132, 11133, 11134, 11135, 11136, where any positive integer, in any word, is <= the sum of the preceding integers.
The subsequence of primes in this partial sum begins: 2, 3, 5, 47, 89, 173, 166289. [From Jonathan Vos Post, Feb 17 2010]
For n > 0: a(n) = A005255(n-1) + 1. - Reinhard Zumkeller, Nov 18 2012

			a(7)=2a(6)-a(3)=2*14-3=25. a(8)=2a(7)-a(3)=2*25-3=47. a(9)=2a(8)-a(4)=2*47-5=89.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

a062178 n = a062178_list !! (n-1)
a062178_list = scanl (+) 0 a002083_list
-- Reinhard Zumkeller, Nov 18 2012

a(n) =a(n-1)+A002083(n).

A155092 Matrix inverse of A155091.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 11, 11, 7, 4, 2, 1, 1, 22, 22, 14, 8, 4, 2, 1, 1, 42, 42, 27, 15, 8, 4, 2, 1, 1, 84, 84, 54, 30, 16, 8, 4, 2, 1, 1, 165, 165, 106, 59, 31, 16, 8, 4, 2, 1, 1, 330, 330, 212, 118, 62, 32, 16, 8, 4, 2, 1, 1, 654, 654, 420, 234, 123, 63

Offset: 1

Mats Granvik, Jan 20 2009

First column is A002083. Rows sums are A045690.

Eigentriangle of A101688. - Paul Barry, Mar 01 2011

			Table begins:
1,
1,1,
1,1,1,
2,2,1,1,
3,3,2,1,1,
6,6,4,2,1,1,
11,11,7,4,2,1,1,
22,22,14,8,4,2,1,1,
42,42,27,15,8,4,2,1,1,
84,84,54,30,16,8,4,2,1,1,

Cf. A002083, A045690.

A174868 Partial sums of Stern's diatomic series A002487.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344, 353, 358, 364, 365, 372, 378, 389, 394, 408, 417, 430, 434, 449, 460, 478, 485, 502, 512, 525, 528, 542, 553, 572, 580, 601, 614, 632, 637, 654, 666, 685

Offset: 0

Jonathan Vos Post, Dec 01 2010

After the initial 0, identical to A007729.

			a(16) = 0 + 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 5 + 2 + 5 + 3 + 4 + 1 = 41.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.
Clemens Heuberger, Daniel Krenn and Gabriel F. Lipnik, Asymptotic Analysis of q-Recursive Sequences, Algorithmica, Vol. 84 (2022), pp. 2480-2532; arXiv preprint, arXiv:2105.04334 [math.CO], 2021-2022.

Cf. A002487, A007729, A084091, A049456, A020946, A020950, A046815, A070871, A070872, A071883, A064881-A064886, A072170, A001045, A002083, A073459, A000123, A126606, A049455.

Cf. A001622, A242208.

a[n_] := a[n] = If[EvenQ[n], 2*a[n/2] + a[n/2 - 1], 2*a[(n - 1)/2] + a[(n + 1)/2]]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, May 18 2023 *)

from itertools import accumulate, count, islice
from functools import reduce
def A174868_gen(): # generator of terms
    return accumulate((sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) for n in count(1)),initial=0)
A174868_list = list(islice(A174868_gen(),30)) # Chai Wah Wu, May 07 2023

a(n) = Sum_{i=0..n} A002487(i).
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 27 2017
a(2k) = 2*a(k) + a(k-1); a(2k+1) = 2*a(k) + a(k+1). - Michael J. Collins, Dec 25 2018
a(n) = n^log_2(3) + Psi_D(log_2(n)) + O(n^log_2(phi)), where phi is the golden ratio (A001622) and Psi_D is a 1-periodic continuous function which is Hölder continuous with any exponent smaller than log_2(3/phi) (Heuberger et al., 2022). - Amiram Eldar, May 18 2023

A005254 Number of weighted voting procedures.

Original entry on oeis.org

1, 3, 9, 21, 51, 117, 271, 607, 1363, 3013, 6643, 14491, 31495, 67965, 146115, 312483, 666015, 1413915, 2992815, 6315135, 13292007, 27906585, 58464339, 122229123, 255072423, 531369483, 1105217223, 2295383319, 4760727375

Offset: 1

N. J. A. Sloane

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 122-123.
T. V. Narayana, Recent progress and unsolved problems in dominance theory, pp. 68-78 of Combinatorial mathematics (Canberra 1977), Lect. Notes Math. Vol. 686, 1978.
T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Solution to Board of Directors Problem, J. Rec. Math., 9 (No. 3, 1977), 240.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
B. E. Wynne & N. J. A. Sloane, Correspondence, 1976-84
B. E. Wynne & T. V. Narayana, Tournament configuration, weighted voting, and partitioned catalans, Preprint.
Bayard Edmund Wynne, and T. V. Narayana, Tournament configuration and weighted voting, Cahiers du bureau universitaire de recherche opérationnelle, 36 (1981): 75-78.
Solution to Board of Directors Problem, J. Rec. Math., 9 (No. 3, 1977), 240. (Annotated scanned copy)

Row sums of A037254.

a005254 = sum . a037254_row  -- Reinhard Zumkeller, Nov 18 2012

a[1, 1] = 1; a[n_, 1] := a[n, 1] = a[n - 1, Floor[(n + 1)/2]]; a[n_, k_ /; k > 1] := a[n, k] = a[n, 1] + a[n - 1, k - 1]; A005254 = Table[ Sum[ a[n, k], {k, 1, n}], {n, 1, 29}] (* Jean-François Alcover, Apr 03 2012, after recurrence of A037254 *)

More terms from James Sellers, Feb 04 2000

A005256 Number of weighted voting procedures.

Original entry on oeis.org

1, 3, 6, 12, 23, 45, 87, 171, 336, 666, 1320, 2628, 5233, 10443, 20841, 41637, 83187, 166287, 332403, 664635, 1328934, 2657532, 5314398, 10628130, 21254940, 42508560, 85014492, 170026356, 340047479, 680089725, 1360169007, 2720327571

Offset: 1

N. J. A. Sloane

T. V. Narayana, Recent progress and unsolved problems in dominance theory, pp. 68-78 of Combinatorial mathematics (Canberra 1977), Lect. Notes Math. Vol. 686, 1978.
T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Herman Jamke, Table of n, a(n) for n = 1..60
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
B. E. Wynne & N. J. A. Sloane, Correspondence, 1976-84
B. E. Wynne & T. V. Narayana, Tournament configuration, weighted voting, and partitioned catalans, Preprint.
Bayard Edmund Wynne, and T. V. Narayana, Tournament configuration and weighted voting, Cahiers du bureau universitaire de recherche opérationnelle, 36 (1981): 75-78.

a005256 n = a005256_list !! (n-1)
a005256_list = map (subtract 2) $ drop 3 a062178_list
-- Reinhard Zumkeller, Nov 18 2012

a[1] = 1; a[2] = 3; a[n_ /; n > 2] := a[n] = 2*a[n-1] - a[Floor[(n-3)/2]]; a[] = 0; Table[a[n], {n, 1, 32}] (* _Jean-François Alcover, Jul 30 2013, after Herman Jamke *)

a(n)=if(n<3, (n>0)+2*(n>1), 2*a(n-1)-a((n-3)\2)) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

a(n+1) = 2*a(n) - a(floor((n-2)/2)) starting with a(1)=1 and a(2)=3 (a(n)=0 if n<1). Also a(n) = A062178(n+2) - 2. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

A043327 a(0)=1; a(1)=1; a(n)= a(n-1) + floor( sqrt(a(n-1)a(n-2))+ sqrt(a(n-3)a(n-4))+ ... ).

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 22, 43, 85, 166, 327, 641, 1259, 2471, 4853, 9528, 18710, 36737, 72137, 141643, 278125, 546112, 1072322, 2105559, 4134379, 8118070, 15940263, 31299549, 61458328, 120676683, 236955071, 465273846, 913589879, 1793882179

Offset: 0

Olivier Gérard, Jul 18 2001

Nonlinear recurrence similar to that for Narayana numbers.

Cf. A002083, A043328.

a[ 0 ]=0; a[ 1 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+ Floor[ Plus@@Map[ Sqrt[ Times@@# ]&, Partition[ Reverse[ Array[ a, n, 0 ] ], 2 ] ] ]

A043328 a(0)=1; a(1)=1; a(n)= a(n-1) + floor(sqrt(a(n-1)a(n-2))) + floor(sqrt(a(n-3)a(n-4))) + ....

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 22, 42, 83, 162, 318, 623, 1224, 2402, 4717, 9261, 18185, 35706, 70111, 137665, 270313, 530772, 1042200, 2046413, 4018242, 7890029, 15492492, 30420327, 59731927, 117286804, 230298862, 452204027, 887926594, 1743490958

Offset: 0

Olivier Gérard, Jul 18 2001

Nonlinear recurrence similar to that for Narayana numbers.

Cf. A002083, A043327.

a[ 0 ]=0; a[ 1 ]=1; a[ 2 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+ Plus@@Map[ Floor[ Sqrt[ Times@@# ] ]&, Partition[ Reverse[ Array[ a, n, 0 ] ], 2 ] ]

A242729 Decimal expansion of the Atkinson-Negro-Santoro constant, a constant associated with Erdős' sum-distinct set constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 21 2014

			0.3166841736527058202183565723...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.28, p. 189.

W. F. Lunnon, Integer sets with distinct subset-sums, Math. Comp. 50 (1988), 297-320.

Cf. A002083, A005255.

digits = 100; Clear[u, s]; u[n_] := u[n] = 2 u[n-1] - u[n-1 - Floor[(n-1)/2 + 1]]; u[0] = 0; u[1] = 1; s[k_] := s[k] = u[k]/2^k // N[#, digits + 5] &; s[dk = 100]; s[k = 2*dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits], Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First

A245892 Number of labeled increasing unary-binary trees on n nodes whose breadth-first reading word simultaneously avoids 231 and 312 (or avoids 312 and 321).

Original entry on oeis.org

1, 1, 3, 8, 28, 102, 407, 1701

Offset: 1

Manda Riehl, Aug 19 2014

The number of labeled increasing unary-binary trees with an associated permutation avoiding 231 and 312 simultaneously (or avoids 312 and 321 simultaneously) in the classical sense. The tree's permutation is found by recording the labels in the order in which they appear in a breadth-first search. (Note that a breadth-first search reading word is equivalent to reading the tree labels left to right by levels, starting with the root.)

In some cases, the same breadth-first search reading permutation can be found on differently shaped trees. This sequence gives the number of trees, not the number of permutations.

			When n=4, a(n)=8.  In the Links above we show the eight labeled increasing trees on four nodes whose permutation simultaneously avoids 231 and 312.

Manda Riehl, The eight trees when n=4.

A111554 gives the number of binary trees instead of unary-binary trees.

A002083 gives the number of permutations which avoid 231 and 312 simultaneously that are breadth-first reading words on labeled increasing unary-binary trees.

cp's OEIS Frontend

A058222 Tree of tournament sequences read across rows.

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A062178 a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1.

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A155092 Matrix inverse of A155091.

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A174868 Partial sums of Stern's diatomic series A002487.

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A005254 Number of weighted voting procedures.

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A005256 Number of weighted voting procedures.

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A043327 a(0)=1; a(1)=1; a(n)= a(n-1) + floor( sqrt(a(n-1)*a(n-2))+ sqrt(a(n-3)*a(n-4))+ ... ).

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A043328 a(0)=1; a(1)=1; a(n)= a(n-1) + floor(sqrt(a(n-1)*a(n-2))) + floor(sqrt(a(n-3)*a(n-4))) + ....

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Mathematica

A242729 Decimal expansion of the Atkinson-Negro-Santoro constant, a constant associated with Erdős' sum-distinct set constant.

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A043327 a(0)=1; a(1)=1; a(n)= a(n-1) + floor( sqrt(a(n-1)a(n-2))+ sqrt(a(n-3)a(n-4))+ ... ).

A043328 a(0)=1; a(1)=1; a(n)= a(n-1) + floor(sqrt(a(n-1)a(n-2))) + floor(sqrt(a(n-3)a(n-4))) + ....