A028399 -id:A028399 - cp's OEIS frontend

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

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

A175164 a(n) = 16*(2^n - 1).

Original entry on oeis.org

0, 16, 48, 112, 240, 496, 1008, 2032, 4080, 8176, 16368, 32752, 65520, 131056, 262128, 524272, 1048560, 2097136, 4194288, 8388592, 16777200, 33554416, 67108848, 134217712, 268435440

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

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

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), this sequence (m=16), A175165 (m=32), A175166 (m=64).

Cf. A140504, A173787.

I:=[0,16]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

16*(2^Range[0,40] - 1) (* G. C. Greubel, Jul 08 2021 *)

def A175164(n): return (1<Chai Wah Wu, Jun 27 2023

[16*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+4) - 16.
a(n) = A173787(n+4, 4).
a(2*n) = A140504(n+2)*A028399(n).
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=16. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 16*x/((1-x)*(1-2*x)).
E.g.f.: 16*(exp(2*x) - exp(x)). (End)

A175166 a(n) = 64*(2^n - 1).

Original entry on oeis.org

0, 64, 192, 448, 960, 1984, 4032, 8128, 16320, 32704, 65472, 131008, 262080, 524224, 1048512, 2097088, 4194240, 8388544, 16777152, 33554368, 67108800, 134217664, 268435392, 536870848, 1073741760

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

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

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), A175165 (m=32), this sequence (m=64).

Cf. A173787, A175161.

I:=[0,64]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

LinearRecurrence[{3,-2},{0,64},30] (* Harvey P. Dale, Apr 08 2015 *)

def A175166(n): return (1<Chai Wah Wu, Jun 27 2023

[64*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+6) - 64.
a(n) = A173787(n+6, 6).
a(2*n) = A175161(n)*A159741(n) for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=64. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 64*x/((1-x)*(1-2*x)).
E.g.f.: 64*(exp(2*x) - exp(x)). (End)

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

A175165 a(n) = 32*(2^n - 1).

Original entry on oeis.org

0, 32, 96, 224, 480, 992, 2016, 4064, 8160, 16352, 32736, 65504, 131040, 262112, 524256, 1048544, 2097120, 4194272, 8388576, 16777184, 33554400, 67108832, 134217696, 268435424, 536870880

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

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

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), this sequence (m=32), A175166 (m=64).

Cf. A173787.

I:=[0,32]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

32(2^Range[0,30] -1) (* or *) LinearRecurrence[{3,-2},{0,32},30] (* Harvey P. Dale, Mar 23 2015 *)

def A175165(n): return (1<Chai Wah Wu, Jun 27 2023

[32*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+5) - 32.
a(n) = A173787(n+5, 5).
a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0, a(1)=32. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 32*x/((1-x)*(1-2*x)).
E.g.f.: 32*(exp(2*x) - exp(x)). (End)

A371064 Array read by ascending antidiagonals where T(n,k) is the number of paths of length k from the origin to a facet of the cross polytope of size k in Z^n.

Original entry on oeis.org

2, 4, 2, 6, 12, 2, 8, 30, 28, 2, 10, 56, 126, 60, 2, 12, 90, 344, 462, 124, 2, 14, 132, 730, 1880, 1566, 252, 2, 16, 182, 1332, 5370, 9368, 5070, 508, 2, 18, 240, 2198, 12372, 36250, 43736, 15966, 1020, 2, 20, 306, 3376, 24710, 106452, 228090, 195224, 49422, 2044, 2

Offset: 1

Shel Kaphan, Mar 09 2024

In the cross polytope of dimension n, each facet of dimension i-1 (i=1..n) has i^k paths of length k from the origin to its surface, and there are binomial(n,i)*2^i such facets. To avoid double counting, an alternating sum is used to add up the paths to all the facets.

			distance
 k      1   2    3      4       5        6         7          8
dims ----------------------------------------------------------
 n 1 |  2   2    2      2       2        2         2          2
   2 |  4  12   28     60     124      252       508       1020
   3 |  6  30  126    462    1566     5070     15966      49422
   4 |  8  56  344   1880    9368    43736    195224     844760
   5 | 10  90  730   5370   36250   228090   1359130    7771770
   6 | 12 132 1332  12372  106452   856212   6505812   47189652
   7 | 14 182 2198  24710  259574  2562182  23928758  213041990
   8 | 16 240 3376  44592  554416  6511920  72592816  772172592

Columns: A002939 (k=2).

Rows: A028399 (n=2), A366058 (n=3).

T(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * 2^i * i^k.

A060158 Number of permutations of [n] with 4 sequences.

Original entry on oeis.org

0, 0, 0, 0, 0, 32, 300, 1852, 9576, 45096, 201060, 866324, 3650592, 15154240, 62260380, 253939116, 1030367448, 4165106264, 16790875860, 67553807428, 271383782544, 1089035545968, 4366631897100, 17497971562460, 70086163646280, 280627369334152, 1123357369925700

Offset: 0

Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Mar 12 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261.

Harry J. Smith, Table of n, a(n) for n = 0..200
E. Rodney Canfield and Herbert S. Wilf, Counting permutations by their runs up and down, arXiv:math/0609704 [math.CO], 2006. [See u_4.]
Index entries for linear recurrences with constant coefficients, signature (13, -67, 175, -244, 172, -48).

Cf. A028399, A060157, A000486, A059427, A000352, A123003.

n4 := n->2*n-7+(6-n)*2^(n-1)-3^n+4^(n-1); seq(n4(i),i=5..27);

Join[{0, 0}, LinearRecurrence[{13, -67, 175, -244, 172, -48}, {0, 0, 0, 32, 300, 1852}, 25]] (* Jean-François Alcover, Sep 02 2018 *)

a(n) = { if (n<2, 0, 2*n - 7 + (6 - n)*2^(n - 1) - 3^n + 4^(n - 1)) } \\ Harry J. Smith, Jul 02 2009

a(n) = 2n - 7 + (6-n)*2^(n-1) - 3^n + 4^(n-1).

G.f.: 4*x^5*(8-29*x+24*x^2)/((1-4*x)*(1-3*x)*(1-2*x)^2*(1-x)^2).

Edited by N. J. A. Sloane, Nov 11 2006

cp's OEIS Frontend

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

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A175164 a(n) = 16*(2^n - 1).

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A175166 a(n) = 64*(2^n - 1).

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

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

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Mathematica

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