A173033 -id:A173033 - cp's OEIS frontend

A137432 Number of ways to place n^2 nonattacking kings on a 2n X 2n cylindrical chessboard.

1, 4, 32, 344, 4460, 66532, 1118398, 20984924, 437500380, 10105541204, 257860425672, 7241521734020, 222770819826574, 7466859257161488, 271156951835070930, 10609740515840572076, 444982726973034212924, 19911203110764903275188, 946564783226311159219150

Offset: 0

Vaclav Kotesovec, Aug 31 2011

Rintaro Matsuo, Table of n, a(n) for n = 0..384 (terms 1..31 from Alex V. Breger)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 209-210.
Artem M. Karavaev, Zealint blog (in Russian)
Rintaro Matsuo, Polynomial-time algorithm
Rintaro Matsuo, a(1)..a(800)

Cf. A018807, A173033.

Conjecture: limit of a(n+1)/(n*a(n)) as n->infinity is e.

a(n) ~ c * n^n, where c = 2*exp(1)*(exp(1) - 1)^2 / (exp(1) - 2)^2 = 31.1116835720490503682643922791052352237386275089... - Vaclav Kotesovec, Jul 29 2023, updated Mar 18 2024

a(11)-a(12) from Vaclav Kotesovec, Sep 08 2011
a(13)-a(27) from Alex V. Breger, Sep 10 2011
a(28)-a(31) from Alex V. Breger, Sep 12 2011
a(0)=1 prepended by Andrew Howroyd, Mar 26 2023

A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

Original entry on oeis.org

0, 1, 3, 3, 7, 2, 15, 4, 9, 5, 31, 3, 63, 2, 8, 16, 127, -1, 255, 4, 21, 16, 511, 8, 21, 20, 12, 27, 1023, 6, 2047, 8, 20, 48, 20, 20, 4095, 2, 78, 32, 8191, -6, 16383, 17, 9, 288, 32767, 8, 45, -3, 122, 45, 65535, 4, 53, 20, 270, 278, 131071, 2, 262143, 688, 12, 72, 56, 23, 524287, 125, 260, -8, 1048575, 20, 2097151, 260, 3, 363, 44, -7, 4194303

Offset: 1

Antti Karttunen, Mar 07 2019

sign

The first four zeros after a(1) occur at n = 192, 288, 3645, 6075.
There are 1562 negative terms among the first 10000 terms.
Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:
----------------------------------------------------------------------------------
  A324201  divisors                             a(n) applied to each:         Sum
        9: [1, 3, 9]                         -> [0, 3, 9]                      12 = 2*6
      125: [1, 5, 25, 125]                   -> [0, 7, 21, 28]                 56 = 2*28
   161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 31, 93, 124, 496, 248]    992 = 2*496
410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]
                             -> [0, 127, 381, 508, 2032, 1016, 9144, 3048]  16256 = 2*8128
The second term (the first nonzero) of the latter list = A000668(n), and the sum is always twice the corresponding perfect number, which forces either it or at least many of its divisors to be present. For example, in the fourth case, although 8128 = A000396(4) itself is not present, we still have 127, 508, 1016 and 2032 in the list. See also A329644.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000040, A000043, A000668, A000203, A000225, A000396, A008683, A068156, A156552, A173033, A297112, A297113, A323243, A323244, A324201, A324542, A324547, A324548, A324549, A324712, A329644.

Table[DivisorSum[n, MoebiusMu[n/#] If[# == 1, 0, DivisorSigma[1, Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]]] &], {n, 79}] (* Michael De Vlieger, Mar 11 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
memoA323243 = Map();
A323243(n) = if(1==n, 0, my(v); if(mapisdefined(memoA323243,n,&v),v, v=sigma(A156552(n)); mapput(memoA323243,n,v); (v)));
A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));

a(n) = Sum_{d|n} A008683(n/d) * A323243(d).
a(A000040(n)) = A000225(n).
a(A001248(n)) = A173033(n) - A000225(n) = A068156(n) = 3*(2^n - 1).
a(2*A000040(n)) = A324549(n).
a(A002110(n)) = A324547(n).
a(n) = 2*A297112(n) - A329644(n), and for n > 1, a(n) = 2^A297113(n) - A329644(n). - Antti Karttunen, Dec 08 2019

A169708 First differences of A169707.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 20, 44, 68, 60, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 20, 44, 68, 76, 84, 156, 196, 140, 84, 156, 212, 236, 324, 508, 516, 252, 4, 12, 20, 28, 20

Offset: 0

N. J. A. Sloane, Apr 17 2010

			From _Omar E. Pol_, Feb 13 2015: (Start)
Written as an irregular triangle in which row lengths are 1,1,2,4,8,16,32,... the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124,20,44,68,76,84,156,196,140,84,156,212,236,324,508,516,252;
It appears that the row sums give A000302.
It appears that the right border gives A173033.
(End)

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000302, A011782, A139251, A151548, A152980, A153006, A160164, A160552, A169707 (partial sums), A170903, A173033, A253088.

It appears that a(n) = 4*A160552(n), n >= 1. - Omar E. Pol, Feb 13 2015

Initial 1 added by Omar E. Pol, Feb 13 2015

A160721 First differences of A160720.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 28, 4, 12, 12, 28, 12, 28, 28, 60, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 12, 28, 28, 60, 28, 60, 60, 124, 28, 60, 60, 124, 60, 124, 124, 252, 4, 12, 12, 28, 12, 28, 28

Offset: 1

Omar E. Pol, May 25 2009, May 29 2009

This sequence is related to the Sierpinski triangle and to Gould's sequence A001316. - Omar E. Pol, Jul 23 2009

When written as a irregular triangle in which row lengths are A011782 it appears that right border gives A173033. - Omar E. Pol, Mar 20 2013

			From _Omar E. Pol_, Mar 20 2013 (Start):
Triangle begins:
1;
4;
4,12;
4,12,12,28;
4,12,12,28,12,28,28,60;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124,12,28,28,60,28,60,60,124,28,60,60,124,60,124,124,252;
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A038573, A139250, A139251, A160720, A160722, A160723.

a(1)=1. Observation: It appears that a(n) = 4*A038573(n-1), n>1. [From Omar E. Pol, Jul 23 2009]. This formula is correct! - N. J. A. Sloane, Jan 23 2016

More terms from R. J. Mathar, Jul 14 2009

A182464 a(n) = 3a(n-1) - 2a(n-2) with a(0)=24 and a(1)=60.

Original entry on oeis.org

24, 60, 132, 276, 564, 1140, 2292, 4596, 9204, 18420, 36852, 73716, 147444, 294900, 589812, 1179636, 2359284, 4718580, 9437172, 18874356, 37748724, 75497460, 150994932, 301989876, 603979764, 1207959540, 2415919092, 4831838196, 9663676404, 19327352820, 38654705652

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 6 vertices.

			a(0) = 6+12+6;
a(1) = 6+12+24+12+6;
a(2) = 6+12+24+48+24+12+6;
a(3) = 6+12+24+48+96+48+24+12+6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182465, A182466, A182467.

CoefficientList[Series[-((12 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2014 *)
LinearRecurrence[{3,-2},{24,60},40] (* Harvey P. Dale, May 27 2018 *)

a(n) = a(n-1)*2 + 12.
a(n) = 12*A153893(n). - Michel Marcus, Jun 01 2014
G.f.: -((12*(x-2))/(2*x^2-3*x+1)). - Vincenzo Librandi, Jun 01 2014

A182467 a(n) = 3a(n-1) - 2a(n-2) with a(0)=36 and a(1)=90.

Original entry on oeis.org

36, 90, 198, 414, 846, 1710, 3438, 6894, 13806, 27630, 55278, 110574, 221166, 442350, 884718, 1769454, 3538926, 7077870, 14155758, 28311534, 56623086, 113246190, 226492398, 452984814, 905969646, 1811939310, 3623878638, 7247757294, 14495514606, 28991029230

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 9 vertices.

			a(0) = 9+18+9;
a(1) = 9+18+36+18+9;
a(2) = 9+18+36+72+36+18+9;
a(3) = 9+18+36+72+144+72+36+18+9.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182465, A182466.

LinearRecurrence[{3,-2},{36,90},30] (* or *) CoefficientList[Series[(-18(x-2))/(1-3x+2x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 29 2013 *)

a(n) = a(n-1)*2 + 18
G.f.: -((18*(x-2))/(2*x^2-3*x+1)). - Harvey P. Dale, Apr 29 2013
a(n) = 18*A153893(n). - Michel Marcus, Jun 01 2014

A182461 a(n) = 3a(n-1) - 2a(n-2) with a(0)=16 and a(1)=40.

Original entry on oeis.org

16, 40, 88, 184, 376, 760, 1528, 3064, 6136, 12280, 24568, 49144, 98296, 196600, 393208, 786424, 1572856, 3145720, 6291448, 12582904, 25165816, 50331640, 100663288, 201326584, 402653176, 805306360, 1610612728, 3221225464, 6442450936, 12884901880

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 4 vertices.

			a(0) = 4+8+4;
a(1) = 4+8+16+8+4;
a(2) = 4+8+16+32+16+8+4;
a(3) = 4+8+16+32+64+32+16+8+4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A000045, A028399, A038578, A089143, A173033, A182462, A182464, A182465, A182466, A182467.

CoefficientList[Series[-((8 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

a(n) = a(n-1)*2 + 8.
G.f.: 16 + 40*x + 88*x^2 + 184*x^3 + 376*x^4 + 760*x^5 + 1528*x^6 + ...
a(n) = 8 * A055010(n+1). [Joerg Arndt, Jun 01 2014]
G.f.: -((8*(x - 2))/(2*x^2 - 3*x + 1)). - Vincenzo Librandi, Jun 02 2014

A182462 a(n) = 3a(n-1) - 2a(n-2) with a(0)=20 and a(1)=50.

Original entry on oeis.org

20, 50, 110, 230, 470, 950, 1910, 3830, 7670, 15350, 30710, 61430, 122870, 245750, 491510, 983030, 1966070, 3932150, 7864310, 15728630, 31457270, 62914550, 125829110, 251658230, 503316470, 1006632950, 2013265910, 4026531830, 8053063670, 16106127350

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 5 vertices.

			a(0) = 5+10+5;
a(1) = 5+10+20+10+5;
a(2) = 5+10+20+40+20+10+5;
a(3) = 5+10+20+40+80+40+20+10+5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182464, A182465, A182466, A182467.

CoefficientList[Series[-((10 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

a(n) = a(n-1)*2 + 10.
a(n) = 10*A153893(n). - Michel Marcus, Jun 01 2014
G.f.: -((10*(x - 2))/(2*x^2 - 3*x + 1)). - Vincenzo Librandi, Jun 02 2014

A182465 a(n) = 3a(n-1) - 2a(n-2) with a(0)=28 and a(1)=70.

Original entry on oeis.org

28, 70, 154, 322, 658, 1330, 2674, 5362, 10738, 21490, 42994, 86002, 172018, 344050, 688114, 1376242, 2752498, 5505010, 11010034, 22020082, 44040178, 88080370, 176160754, 352321522, 704643058, 1409286130, 2818572274, 5637144562, 11274289138, 22548578290

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 7 vertices.

			a(0) = 7+14+7;
a(0) = 7+14+28+14+7;
a(0) = 7+14+28+56+28+14+7;
a(0) = 7+14+28+56+112+56+28+14+7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182466, A182467.

CoefficientList[Series[-((14 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2014 *)
LinearRecurrence[{3,-2},{28,70},30] (* Harvey P. Dale, Oct 05 2015 *)

a(n) = a(n-1)*2 + 14.
a(n) = 14*A153893(n). - Michel Marcus, Jun 01 2014
G.f.: -((14*(x-2))/(2*x^2-3*x+1)). - Vincenzo Librandi, Jun 01 2014

A182466 a(n) = 3a(n-1) - 2a(n-2) with a(0)=32 and a(1)=80.

Original entry on oeis.org

32, 80, 176, 368, 752, 1520, 3056, 6128, 12272, 24560, 49136, 98288, 196592, 393200, 786416, 1572848, 3145712, 6291440, 12582896, 25165808, 50331632, 100663280, 201326576, 402653168, 805306352, 1610612720, 3221225456, 6442450928, 12884901872, 25769803760, 51539607536

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 8 vertices.

			a(0) = 8+16+8;
a(1) = 8+16+32+16+8;
a(2) = 8+16+32+64+32+16+8;
a(3) = 8+16+32+64+128+64+32+16+8.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182465, A182467.

LinearRecurrence[{3,-2},{32,80},40] (* or *) Table[8(3*2^n-2),{n,40}] (* Harvey P. Dale, Aug 23 2012 *)
CoefficientList[Series[-((16 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

a(n) = a(n-1)*2 + 16.
a(n) = 8*(3*2^n-2). - Harvey P. Dale, Aug 23 2012
G.f.: -((16(x-2))/(2*x^2-3*x+1)). - Harvey P. Dale, Aug 23 2012

cp's OEIS Frontend

A137432 Number of ways to place n^2 nonattacking kings on a 2n X 2n cylindrical chessboard.

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A324543 Möbius transform of A323243, where A323243(n) = sigma(A156552(n)).

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A169708 First differences of A169707.

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A160721 First differences of A160720.

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A182464 a(n) = 3a(n-1) - 2a(n-2) with a(0)=24 and a(1)=60.

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A182467 a(n) = 3a(n-1) - 2a(n-2) with a(0)=36 and a(1)=90.

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A182461 a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=16 and a(1)=40.

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A182462 a(n) = 3a(n-1) - 2a(n-2) with a(0)=20 and a(1)=50.

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A182461 a(n) = 3a(n-1) - 2a(n-2) with a(0)=16 and a(1)=40.