A016129 -id:A016129 - cp's OEIS frontend

A093050 Exponent of 2 in (3^n-3)*2^(n-1).

0, 0, 3, 2, 6, 4, 7, 6, 11, 8, 11, 10, 14, 12, 15, 14, 20, 16, 19, 18, 22, 20, 23, 22, 27, 24, 27, 26, 30, 28, 31, 30, 37, 32, 35, 34, 38, 36, 39, 38, 43, 40, 43, 42, 46, 44, 47, 46, 52, 48, 51, 50, 54, 52, 55, 54, 59, 56, 59, 58, 62, 60, 63, 62, 70, 64, 67, 66, 70

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

Cf. A090740, A093051, A093052.

a(n) is the exponent of 2 in A016129(n-1), A024281(n), A024287(n), A066406(n)/2, A071952(n+3).

a(n)=if(n<1,0,if(n%2==0,a(n/2)+2*floor((n+2)/4)+1,n-1))

Recurrence: a(2n) = a(n) + [(n+1)/2] + 1, a(2n+1) = 2n.
G.f.: Sum_{k>=0} t^2(3+2t+2t^3-t^4)/[(1+t^2)(1-t^2)^2], t=x^2^k.
a(n) = A093051(n) - 1 = A090740(n) + n - 2, for n >= 1. - Amiram Eldar, Sep 14 2024

A165147 a(n) = (3*7^n-3^n)/2.

Original entry on oeis.org

1, 9, 69, 501, 3561, 25089, 176109, 1234221, 8643921, 60520569, 423683349, 2965901541, 20761665081, 145332718449, 1017332217789, 7121335090461, 49849374331041, 348945706410729, 2442620203155429, 17098342196928981

Offset: 0

Klaus Brockhaus, Sep 15 2009

Partial sums are in A016138.

Binomial transform of A016129. Inverse binomial transform of A165148.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A016138, A016129, A165148.

Magma
```
[ (3*7^n-3^n)/2: n in [0..19] ];
```

LinearRecurrence[{10, -21}, {1, 9}, 25] (* Paolo Xausa, Apr 22 2024 *)
Table[(3*7^n-3^n)/2,{n,0,20}] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = 10*a(n-1)-21*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

G.f.: (1-x)/((1-3*x)*(1-7*x)).

A165152 a(n) = (3*12^n - 8^n)/2.

Original entry on oeis.org

1, 14, 184, 2336, 29056, 356864, 4347904, 52699136, 636583936, 7672561664, 92339175424, 1110217588736, 13339790934016, 160214930161664, 1923678673567744, 23092940175835136, 277185650854199296, 3326790760203812864, 39925992722073124864, 479147941461896462336

Offset: 0

Klaus Brockhaus, Sep 15 2009

Binomial transform of A165151. Sixth binomial transform of A016129. Tenth binomial transform of A056450.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (20,-96).

Cf. A165151, A016129, A056450.

Magma
```
[ (3*12^n-8^n)/2: n in [0..17] ];
```

A165152:=n->(3*12^n - 8^n)/2; seq(A165152(n), n=0..17); # Wesley Ivan Hurt, Jan 29 2014

Table[(3*12^n - 8^n)/2, {n, 0, 17}] (* Wesley Ivan Hurt, Jan 29 2014 *)

a(n) = 20*a(n-1) - 96*a(n-2) for n > 1; a(0) = 1, a(1) = 14.
G.f.: (1 - 6*x)/((1 - 8*x)*(1 - 12*x)).
E.g.f.: exp(8*x)*(3*exp(4*x) - 1)/2. - Stefano Spezia, Mar 30 2023

A241275 a(n) = 6a(n-1) + 5(2^(n-1)-1) for n > 0, a(0) = 0.

Original entry on oeis.org

0, 0, 5, 45, 305, 1905, 11585, 69825, 419585, 2518785, 15115265, 90696705, 544190465, 3265163265, 19591020545, 117546205185, 705277394945, 4231664697345, 25389988839425, 152339934347265, 914039608705025, 5484237657473025, 32905425955323905, 197432555752914945, 1184595334559432705, 7107572007440482305

Offset: 0

Kival Ngaokrajang, Apr 18 2014

a(n) is the total number of holes in a pentaflake after n iterations. The number of irregular polygon-shaped holes is 5*A016200(n-3), n >= 3. The number of rhombic-shaped holes is 5*A016129(n-2), n >= 2 and equal to the first difference of a(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A016129, A016200, A240523.

A241275:=n->9*6^(n-2) - 5*2^(n-2) + 1; seq(A241275(n), n=0..40); # Wesley Ivan Hurt, Apr 19 2014

CoefficientList[Series[-5 x^2/((x - 1) (2 x - 1) (6 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)
LinearRecurrence[{9,-20,12},{0,0,5},30] (* Harvey P. Dale, Oct 10 2017 *)

a(n)=if(n<=0, 0, a(n-1)*6+5*(2^(n-1)-1))
       for(n=0, 100, print1(a(n),", "))

a(n) = 5*(A016129(n-2) + A016200(n-3)) for n >= 3, a(0)=a(1)=0, a(2)=5.
a(n) = 5*A016200(n-2) for n >= 2.
a(n) = 9*6^(n-2) - 5*2^(n-2) + 1.
a(n) = (4-5*2^n+6^n)/4. G.f.: -5*x^2 / ((x-1)*(2*x-1)*(6*x-1)). - Colin Barker, Apr 18 2014

Typo in DATA fixed by Colin Barker, Apr 18 2014

A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 4, 4, 0, 0, 1, 6, 14, 8, 0, 0, 1, 8, 30, 48, 16, 0, 0, 1, 10, 52, 144, 164, 32, 0, 0, 1, 12, 80, 320, 684, 560, 64, 0, 0, 1, 14, 114, 600, 1936, 3240, 1912, 128, 0, 0, 1, 16, 154, 1008, 4400, 11648, 15336, 6528, 256, 0

Offset: 0

Seiichi Manyama, Mar 11 2023

			Square array begins:
  0,  0,   0,   0,    0,    0, ...
  1,  1,   1,   1,    1 ,   1, ...
  0,  2,   4,   6,    8,   10, ...
  0,  4,  14,  30,   52,   80, ...
  0,  8,  48, 144,  320,  600, ...
  0, 16, 164, 684, 1936, 4400, ...

Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.
Main diagonal gives A360766.
Cf. A361432.

T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);

T(0,k) = 0, T(1,k) = 1; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n - (k - sqrt(k))^n)/(2 * sqrt(k)) for k > 0.
G.f. of column k: x/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 11, 83, 535, 3171, 17871, 97483, 520055, 2731091, 14179231, 72992283, 373347975, 1900290211, 9635660591, 48715157483, 245723238295, 1237206060531, 6220389909951, 31239388241083, 156746696495015, 785932504682051, 3938458614335311, 19727477439571083

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-38,40).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

CoefficientList[ Series[ 1/((1 - 2x)(1 - 4x)(1 - 5x)), {x, 0, 20} ], x ]
LinearRecurrence[{11,-38,40},{1,11,83},30] (* Harvey P. Dale, Nov 29 2022 *)

Vec(1/((1-2*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(5^n - 2^n)/3-(4^n - 2^n)/2 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (2/3)*2^n - 8*(4)^n + (25/3)*5^n. - Antonio Alberto Olivares, May 12 2012

A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 13, 117, 905, 6461, 43953, 289717, 1868425, 11861421, 74423393, 462815717, 2858273145, 17556537181, 107373722833, 654414852117, 3977351721065, 24118423433741, 145982106270273, 882250466222917

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-52,60).

Cf. A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{13,-52,60},{1,13,117},20] (* Harvey P. Dale, Mar 26 2016 *)

[(6^n - 2^n)/4-(5^n - 2^n)/3 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = A016129(n+1) - A016127(n+1). - Zerinvary Lajos, Jun 05 2009
a(n) = 13*a(n-1) - 52*a(n-2) + 60*a(n-3), n >= 3.
a(n) = 11*a(n-1) - 30*a(n-2) + 2^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 7*a(n-1) - 10*a(n-2) + 6^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 8*a(n-1) - 12*a(n-2) + 5^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = -5^(n+2)/3 + 9*6^n + 2^n/3. - R. J. Mathar, Mar 18 2011

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 25, 447, 6989, 101759, 1417941, 19180519, 253983853, 3309800367, 42599540357, 542895780791, 6863463633117, 86197420501375, 1076563471968373, 13382900349107463, 165700329729679181, 2044564737700501583, 25152545442794015589, 308625999807796411735, 3778261997130507936445

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (25,-178,264).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

Cf. A016135, A016136.

[(648*12^n +2^(n+1)-5*11^(n+2))/45 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 11 x) (1 - 12 x)), {x, 0, 15}], x] (* Michael De Vlieger, Jan 31 2018 *)

Vec(1/((1-2*x)*(1-11*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(12^n - 2^n)/10-(11^n - 2^n)/9 for n in range(2,18)] # Zerinvary Lajos, Jun 05 2009

From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (648*12^n + 2^(n+1) - 5*11^(n+2))/45.
a(n) = 23*a(n-1) - 132*a(n-2) + 2^n.
a(n) = 25*a(n-1) - 178*a(n-2) + 264*a(n-3), n >= 3. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(2*x)*(648*exp(10*x) - 605*exp(9*x) + 2)/45.
a(n) = A016136(n+1) - A016135(n+1). (End)

A096039 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^5-M)/4, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

Original entry on oeis.org

1, 6, 2, 31, 18, 3, 156, 124, 36, 4, 781, 780, 310, 60, 5, 3906, 4686, 2340, 620, 90, 6, 19531, 27342, 16401, 5460, 1085, 126, 7, 97656, 156248, 109368, 43736, 10920, 1736, 168, 8, 488281, 878904, 703116, 328104, 98406, 19656, 2604, 216, 9, 2441406

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle begins:
1;
6,       2;
31,     18,    3;
156,   124,   36,   4;
781,   780,  310,  60,  5;
3906, 4686, 2340, 620, 90, 6;

Cf. A007318. First column gives A003463. Row sums give A016129.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^5-M)/4 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Oct 07 2009

max = 11; M = Table[If[k > n, 0, Binomial[n, k]], {n, 0, max}, {k, 0, max} ];
T = (MatrixPower[M, 5] - M)/4;
Table[T[[n + 1]][[1 ;; n]] , {n, 1, max}] // Flatten (* Jean-François Alcover, May 24 2016 *)

Edited with more terms by Alois P. Heinz, Oct 07 2009

A370762 Triangle read by rows: T(n,k) = 2 * (k mod 2 + 1) * T(n-1,floor(k/2)) + 1 with T(0,0) = 1 for 0 <= k <= 2^n-1.

Original entry on oeis.org

1, 3, 5, 7, 13, 11, 21, 15, 29, 27, 53, 23, 45, 43, 85, 31, 61, 59, 117, 55, 109, 107, 213, 47, 93, 91, 181, 87, 173, 171, 341, 63, 125, 123, 245, 119, 237, 235, 469, 111, 221, 219, 437, 215, 429, 427, 853, 95, 189, 187, 373, 183, 365, 363, 725, 175, 349, 347, 693, 343, 685, 683, 1365

Offset: 0

Seiichi Manyama, Mar 01 2024

			First few rows are:
   1;
   3,  5;
   7, 13, 11,  21;
  15, 29, 27,  53, 23,  45,  43,  85;
  31, 61, 59, 117, 55, 109, 107, 213, 47, 93, 91, 181, 87, 173, 171, 341;

Seiichi Manyama, Rows n = 0..12, flattened

Row sums give A016129.
Columns k=0 gives A000225(n+1).
Cf. A247648.

T(n, k) = if(n==0, 1, 2*(k%2+1)*T(n-1, k\2)+1);

cp's OEIS Frontend

A093050 Exponent of 2 in (3^n-3)*2^(n-1).

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A165147 a(n) = (3*7^n-3^n)/2.

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Magma

Mathematica

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A165152 a(n) = (3*12^n - 8^n)/2.

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Magma

Maple

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A241275 a(n) = 6*a(n-1) + 5*(2^(n-1)-1) for n > 0, a(0) = 0.

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A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

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A016282 Expansion of 1/((1-2*x)*(1-4*x)*(1-5*x)).

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Mathematica

PARI

Sage

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A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

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Mathematica

Sage

Formula

A016633 Expansion of g.f. 1/((1-2*x)*(1-11*x)*(1-12*x)).

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A241275 a(n) = 6a(n-1) + 5(2^(n-1)-1) for n > 0, a(0) = 0.

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).