A222057 -id:A222057 - cp's OEIS frontend

A068156 G.f.: (x+2)(x+1)/((x-1)(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.

1, 3, 9, 21, 45, 93, 189, 381, 765, 1533, 3069, 6141, 12285, 24573, 49149, 98301, 196605, 393213, 786429, 1572861, 3145725, 6291453, 12582909, 25165821, 50331645, 100663293, 201326589, 402653181, 805306365, 1610612733

Offset: 0

Benoit Cloitre, Mar 12 2002

Number of moves to solve Hard Pagoda puzzle.
Partial sums of A111286. Binomial transform of (1,2,4,2,4,2,4 ....). - Paul Barry, Feb 28 2003
Warren W. Kokko writes that this sequence also appears to give the number of scoring sequences for the Racer Dice Game with n dice. - N. J. A. Sloane, Feb 24 2015
From Michel Lagneau, Apr 27 2015: (Start)
For n > 0, a(n) is the number of identical bowls having the same weight except for one which has a higher weight than the others which are identifiable by a weighing machine using n weighings.
Example: a(2)=9 because two weighings are sufficient:
Start with 9 bowls;
Step 1: remove 3 bowls => there are still 6 bowls;
Step 2: first weighing of 6 bowls (3 bowls on each side of the weighing machine);
Step 3: if the machine is in equilibrium, we find immediately the unknown bowl with a second weighing from the first 3 removing bowls. Else, we find immediately the unknown bowl with a second weighing from the 3 heaviest bowls.
Note: If the unknown bowl has a lower weight, the reasoning is the same, but it is necessary to know whether the unknown bowl is heavier or lighter.
In the general case, we always remove 3 bowls in step 1.
(End)
The number of ternary words of length n that avoid {11-2,22-1}. G.f. [1+(k-1)*x^2]/[1-k*x+(k-1)*x^2] at k=3. [Theorem 7.93 by Heubach and Mansour]. - R. J. Mathar, May 22 2016
Apart from the first term, column 2 of A222057. - Anton Zakharov, Oct 27 2016

Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.
Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.
S. Heubach, T. Mansour, in Combinatorics of Compositions and words, Discr. Math. Applicat. (ed by K H Rosen), CRC Press 2010, p 300.
Warren W. Kokko, The Racer Dice Game, Manuscript, 2015.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Artur Schaefer, Endomorphisms of The Hamming Graph and Related Graphs, arXiv preprint arXiv:1602.02186 [math.CO], 2016. See Table Remark 4.5.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

A diagonal of A233308 (for n > 1).

Cf. A000079.

[3*2^n+0^n-3 : n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

Join[{1}, LinearRecurrence[{3, -2}, {3, 9}, 30]] (* Jean-François Alcover, Jan 08 2019 *)
CoefficientList[Series[(1+2x^2)/((1-2x)(1-x)),{x,0,40}],x] (* Harvey P. Dale, Jan 02 2022 *)

def a(n): return 3*2**n+0**n-3 # Torlach Rush, Jan 09 2025

a(0) = 1, a(n) = A060482(2n+1). For n > 0, a(n+1) = 2*a(n)+3.
G.f.: (1+2*x^2)/((1-2*x)*(1-x)). - Paul Barry, Feb 28 2003
a(n) = 3*2^n+0^n-3. - Paul Barry, Sep 04 2003
a(n) = A099257(A033484(n)+1) = 2*A033484(n) + 1. - Reinhard Zumkeller, Oct 09 2004
a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - Vincenzo Librandi, Nov 11 2011
a(n) = a(n-1)+ 3*2^(n-1); a(1)=3. - Ctibor O. Zizka, Apr 17 2008
E.g.f.: 1 + 3*(exp(x) - 1)*exp(x). - Ilya Gutkovskiy, May 22 2016

A222058 Harmonic-geometric numbers.

Original entry on oeis.org

0, 1, 4, 21, 138, 1095, 10208, 109473, 1328470, 18003675, 269580492, 4420677525, 78801184322, 1517300654415, 31386251780536, 694190761402377, 16348768018619694, 408472183061464515, 10791720442056792740, 300605598797790229629, 8805117712245004098586, 270562051319419652165175, 8702576800277309526639504, 292425620801795849417200881

Offset: 0

N. J. A. Sloane, Feb 08 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Row sums of A222057 or A222060.

Cf. A000254.

Table[Sum[StirlingS2[n, k]*Abs[StirlingS1[k + 1, 2]], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2013 *)

a(n) = Sum_{k=0..n} Stirling2(n,k)*|Stirling1(k+1,2)|.
Maximal term in the sum is asymptotically in position k = n/(2*log(2)) and limit n-> infinity (a(n)/n!)^(1/n) = 1/log(2). - Vaclav Kotesovec, Feb 09 2013
E.g.f.: -log(2 - exp(x))/(2 - exp(x)). - Ilya Gutkovskiy, May 31 2018
a(n) ~ n! * log(n) / (2 * (log(2))^(n+1)) * (1 + (gamma - log(2) - log(log(2))) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 13 2018

A222064 a(n) = n-th third-order hypergeometric-harmonic number.

Original entry on oeis.org

0, 1, 8, 69, 674, 7455, 92540, 1276569, 19394870, 321982323, 5801055632, 112753640109, 2352074473226, 52419496769991, 1243115350746404, 31257697673933889, 830700701852539742, 23266435856618600859, 684997785857198880056, 21149644833172896698709

Offset: 0

N. J. A. Sloane, Feb 08 2013

nonn

Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Cf. A222057-A222064. Row sums of A222063.

hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
a(n) = {sum(k=0, n, k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,3));}
\\ Michel Marcus, Feb 09 2013

a(n) = Sum_{k=0..n} A008277(n,k)*A000142(k)*H3(k) where H3(k) is defined by g.f.:- log(1-x)/(1-x)^3. - Michel Marcus, Feb 09 2013

More terms from Michel Marcus, Feb 09 2013

A222060 Triangle read by rows: coefficients of harmonic-geometric polynomials.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 9, 11, 0, 1, 21, 66, 50, 0, 1, 45, 275, 500, 274, 0, 1, 93, 990, 3250, 4110, 1764, 0, 1, 189, 3311, 17500, 38360, 37044, 13068, 0, 1, 381, 10626, 85050, 287700, 469224, 365904, 109584, 0, 1, 765, 33275, 388500, 1904574, 4667544, 6037416, 3945024, 1026576, 0, 1, 1533, 102630, 1705250, 11651850, 40266828, 76839840, 82188000, 46195920, 10628640

Offset: 0

N. J. A. Sloane, Feb 08 2013

			Triangle begins:
[0]
[0, 1]
[0, 1, 3]
[0, 1, 9, 11]
[0, 1, 21, 66, 50]
[0, 1, 45, 275, 500, 274]
[0, 1, 93, 990, 3250, 4110, 1764]
[0, 1, 189, 3311, 17500, 38360, 37044, 13068]
[0, 1, 381, 10626, 85050, 287700, 469224, 365904, 109584]
...

Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Row sums give A222058. See A222057 for another version.

The n-th polynomial is Sum_{k=0..n} Stirling2(n,k)*|Stirling1(k+1,2)|*x^k.

A222061 Triangle read by rows: coefficients of second-order hypergeometric-harmonic polynomials.

Original entry on oeis.org

0, 0, 1, 0, 1, 5, 0, 1, 15, 26, 0, 1, 35, 156, 154, 0, 1, 75, 650, 1540, 1044, 0, 1, 155, 2340, 10010, 15660, 8028, 0, 1, 315, 7826, 53900, 146160, 168588, 69264, 0, 1, 635, 25116, 261954, 1096200, 2135448, 1939392, 663696, 0, 1, 1275, 78650, 1196580, 7256844

Offset: 0

N. J. A. Sloane, Feb 08 2013

			Triangle begins:
  0
  0 1
  0 1 5
  0 1 15 26
  0 1 35 156 154
  0 1 75 650 1540 1044
  ....

Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Cf. A222057-A222064. Row sums are in A222062.

H2[k_] := (k+1) (HarmonicNumber[k+1] - 1);
T[n_, k_] := StirlingS2[n, k] k! H2[k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 01 2018 *)

hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2)};
\\ Michel Marcus, Feb 09 2013

T(n,k) = A008277(n,k)*A000142(k)*H2(k) where H2(k) is defined by g.f.:- log(1-x)/(1-x)^2. - Michel Marcus, Feb 09 2013

More terms from Michel Marcus, Feb 09 2013

A222062 a(n) = n-th second-order hypergeometric-harmonic number.

Original entry on oeis.org

0, 1, 6, 42, 346, 3310, 36194, 446054, 6122442, 92668302, 1533812722, 27565147126, 534621745178, 11131104732254, 247646911102530, 5863652049020358, 147225092025474154, 3907328980930705966, 109297865960259305618, 3214017757399205062550, 99121172016580291190970

Offset: 0

N. J. A. Sloane, Feb 08 2013

nonn

Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Cf. A222057-A222064. Row sums of A222061.

hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
a(n) = {sum(k=0, n, k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2));}
\\ Michel Marcus, Feb 09 2013

a(n) = Sum_{k=0..n} A008277(n,k)*A000142(k)*H2(k) where H2(k) is defined by g.f.: - log(1-x)/(1-x)^2. - Michel Marcus, Feb 09 2013

More terms from Michel Marcus, Feb 09 2013

A222063 Triangle read by rows: coefficients of third-order hypergeometric-harmonic polynomials.

Original entry on oeis.org

0, 0, 1, 0, 1, 7, 0, 1, 21, 47, 0, 1, 49, 282, 342, 0, 1, 105, 1175, 3420, 2754, 0, 1, 217, 4230, 22230, 41310, 24552, 0, 1, 441, 14147, 119700, 385560, 515592, 241128, 0, 1, 889, 45402, 581742, 2891700, 6530832, 6751584, 2592720, 0, 1, 1785, 142175, 2657340

Offset: 0

N. J. A. Sloane, Feb 08 2013

			Triangle begins:
0
0 1
0 1 7
0 1 21 47
0 1 49 282 342
0 1 105 1175 3420 2754
....

Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Cf. A222057-A222064. Row sums give A222064.

hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,3)};
\\ Michel Marcus, Feb 09 2013

T(n,k) = A008277(n,k)*A000142(k)*H3(k) where H3(k) is defined by g.f.:- log(1-x)/(1-x)^3. - Michel Marcus, Feb 09 2013

More terms from Michel Marcus, Feb 09 2013

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A068156 G.f.: (x+2)*(x+1)/((x-1)*(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.

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A222058 Harmonic-geometric numbers.

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A222064 a(n) = n-th third-order hypergeometric-harmonic number.

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A222060 Triangle read by rows: coefficients of harmonic-geometric polynomials.

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A222061 Triangle read by rows: coefficients of second-order hypergeometric-harmonic polynomials.

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A222062 a(n) = n-th second-order hypergeometric-harmonic number.

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A222063 Triangle read by rows: coefficients of third-order hypergeometric-harmonic polynomials.

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PARI

Formula

Extensions

A068156 G.f.: (x+2)(x+1)/((x-1)(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.