A005061 -id:A005061 - cp's OEIS frontend

A248225 a(n) = 6^n - 3^n.

0, 3, 27, 189, 1215, 7533, 45927, 277749, 1673055, 10058013, 60407127, 362619909, 2176250895, 13059099693, 78359381127, 470170635669, 2821066860735, 16926530304573, 101559569247927, 609358577749029, 3656154953278575, 21936940180024653

Offset: 0

Vincenzo Librandi, Oct 04 2014

Index entries for linear recurrences with constant coefficients, signature (9,-18).

Cf. sequences of the form k^n-3^n: A005061 (k=4), A005058 (k=5), this sequence (k=6), A190541 (k=7), A190543 (k=8), A059410 (k=9), A248226 (k=10), A139741 (k=11).

Cf. A000225, A000244.

Magma
```
[6^n-3^n: n in [0..30]];
```

Table[6^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[3 x / ((1 - 3 x) (1 - 6 x)), {x, 0, 30}], x]
LinearRecurrence[{9,-18},{0,3},30] (* Harvey P. Dale, Jul 12 2025 *)

G.f.: 3*x/((1-3*x)*(1-6*x)).
a(n) = 9*a(n-1) - 18*a(n-2).
a(n) = 3^n*(2^n - 1) = A000244(n)*A000225(n).
E.g.f.: 2*exp(9*x/2)*sinh(3*x/2). - Elmo R. Oliveira, Mar 31 2025

A327318 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 1 and s = 1/2.

Original entry on oeis.org

1, 3, 4, 7, 18, 12, 15, 56, 72, 32, 31, 150, 280, 240, 80, 63, 372, 900, 1120, 720, 192, 127, 882, 2604, 4200, 3920, 2016, 448, 255, 2032, 7056, 13888, 16800, 12544, 5376, 1024, 511, 4590, 18288, 42336, 62496, 60480, 37632, 13824, 2304, 1023, 10220, 45900

Offset: 1

Clark Kimberling, Nov 08 2019

p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			First six rows:
   1;
   3,   4;
   7,  18,  12;
  15,  56,  72,   32;
  31, 150, 280,  240,  80;
  63, 372, 900, 1120, 720, 192;
The first six polynomials, not factored:
1, 3 + 4 x, 7 + 18 x + 12 x^2, 15 + 56 x + 72 x^2 + 32 x^3, 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, 63 + 372 x + 900 x^2 + 1120 x^3 + 720 x^4 + 192 x^5.
The first six polynomials, factored:
1, 3 + 4 x, 7 + 18 x + 12 x^2, (3 + 4 x) (5 + 12 x + 8 x^2), 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, (3 + 4 x) (3 + 6 x + 4 x^2) (7 + 18 x + 12 x^2).

Cf. A327316, A327317, A000225 (x = 0), A005061 (x = 1), A081199 (x = 1/2).

r = 1; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327318 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327318 sequence *)

A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 7, 10, 10, 7, 15, 22, 24, 22, 15, 31, 46, 52, 52, 46, 31, 63, 94, 108, 112, 108, 94, 63, 127, 190, 220, 232, 232, 220, 190, 127, 255, 382, 444, 472, 480, 472, 444, 382, 255, 511, 766, 892, 952, 976, 976, 952, 892, 766, 511, 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

T(n, k) is the expanded number of player-reduced static games within an n-player two-strategy game scenario in which one player (the "standpoint") faces a specific combination of other players' individual strategies with the possibility of anti-coordination between them -- the total number of such combinations is 2^(n-1). The value of k represents the number of other players who (are expected to) agree on one of the two strategies in S, while the other n-k-1 choose the other strategy; the standpoint player is not included.

			Triangle begins:
  0;
  1,     1;
  3,     4,    3;
  7,    10,   10,    7;
  15,   22,   24,   22,   15;
  31,   46,   52,   52,   46,   31;
  63,   94,  108,  112,  108,   94,   63;
 127,  190,  220,  232,  232,  220,  190,  127;
 255,  382,  444,  472,  480,  472,  444,  382,  255;
 511,  766,  892,  952,  976,  976,  952,  892,  766,  511;
1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
  ...

Column k=0 gives A000225(n-1).
Column k=1 gives A033484(n-2).
Column k=2 gives A053208(n-3).
Cf. A005061, A359200.

T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
seq(T(n), n=1..12);

T[n_, k_] := 2^n - 2^(n - k - 1) - 2^k; Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

T(n, k) = 2^n - 2^(n-k-1) - 2^k.

Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)

A383755 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 3^(n-k) * T(n-1,k-1) + 4^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 37, 37, 1, 1, 175, 925, 175, 1, 1, 781, 19525, 19525, 781, 1, 1, 3367, 375661, 1776775, 375661, 3367, 1, 1, 14197, 6828757, 144142141, 144142141, 6828757, 14197, 1, 1, 58975, 119609725, 10884484975, 48575901517, 10884484975, 119609725, 58975, 1

Offset: 0

Seiichi Manyama, May 09 2025

			Triangle begins:
  1;
  1,    1;
  1,    7,      1;
  1,   37,     37,       1;
  1,  175,    925,     175,      1;
  1,  781,  19525,   19525,    781,    1;
  1, 3367, 375661, 1776775, 375661, 3367, 1;
  ...

Columns k=0..3 give A000012, A005061, A383756(n-2), A383757(n-3).

Cf. A022168.

T(n, k) = if(n*k==0, n^k, 3^(n-k)*T(n-1, k-1)+4^k*T(n-1, k));

def a_row(n): return [3^(k*(n-k))*q_binomial(n, k, 4/3) for k in (0..n)]
for n in (0..8): print(a_row(n))

T(n,k) = 3^(k*(n-k)) * q-binomial(n, k, 4/3).
T(n,k) = 4^(n-k) * T(n-1,k-1) + 3^k * T(n-1,k).
T(n,k) = T(n,n-k).
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 3^j.

A065622 Numerator of 1 - (3/4)^n - frac((3/2)^n), where frac(x) = x - floor(x).

Original entry on oeis.org

0, -1, 3, 13, 159, 173, 1767, 12789, 17759, 126237, 292183, 1930245, 3724303, 23940141, 14206087, 99585429, 640559295, 12562430525, 7042526903, 43417422885, 813747135599, 494896655693, 3000760993767, 18098709141429, 249612172740383

Offset: 0

Henry Bottomley, Dec 03 2001

The presumption that the fraction is positive for n > 1 underlies the presumed solution to Waring's problem.

			a(3) = 13 since 1 - (3/4)^3 - frac((3/2)^3) = 1 - 27/64 - frac(27/8) = 1 - 27/64 - 3/8 = (64 - 27 - 24)/64 = 13/64.

Harry J. Smith, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Waring's Problem

Denominator is A000302. Cf. A002804.

Table[1 - (3/4)^n - FractionalPart[(3/2)^n], {n, 0, 24}] // Numerator (* Jean-François Alcover, Apr 26 2016 *)

{ for (n=0, 200, a=numerator(1 - (3/4)^n - frac((3/2)^n)); write("b065622.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 24 2009

a(n) = 4^n*(1 + floor((3/2)^n)) - 3^n - 6^n = A005061(n) - A002380(n)*A000079(n) = A000302(n)*(1 + A002379(n)) - A000244(n) - A000400(n).

A065814 a(n) = tau(n)^2 - tau(n^2), where tau(n) = A000005(n).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 1, 9, 4, 7, 1, 21, 1, 7, 7, 16, 1, 21, 1, 21, 7, 7, 1, 43, 4, 7, 9, 21, 1, 37, 1, 25, 7, 7, 7, 56, 1, 7, 7, 43, 1, 37, 1, 21, 21, 7, 1, 73, 4, 21, 7, 21, 1, 43, 7, 43, 7, 7, 1, 99, 1, 7, 21, 36, 7, 37, 1, 21, 7, 37, 1, 109, 1, 7, 21, 21, 7, 37, 1, 73, 16, 7, 1, 99, 7, 7, 7, 43

Offset: 1

Labos Elemer, Nov 22 2001

nonn

If n = p^c = power of a prime, then a(n) = (c+1)^2 - (2c+1) = c^2. If n is squarefree with k prime factors then a(n) = 4^k - 3^k, so A065814(A002110(n)) = 4^n - 3^n = A005061(n). Terms depend on prime signature only.

If n is a prime (A000040), then a(n) = 1. If n is a semiprime (A001358), then a(n) = 4 + 3*ceiling(sqrt(n)) - 3*floor(sqrt(n)). If n is a triprime (A014612), then a(n) = 9 * floor(1/omega(n)) + 21 * (1 - (omega(n) mod 2)) + 37 * floor(omega(n)/3), n > 1. - Wesley Ivan Hurt, May 24 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A000040, A000290, A000961, A001358, A002110, A005061, A014612, A035116, A048691.

a[n_] := DivisorSigma[0, n]^2 - DivisorSigma[0, n^2]; Array[a, 100] (* Amiram Eldar, Apr 25 2024 *)

a(n) = { numdiv(n)^2 - numdiv(n^2) } \\ Harry J. Smith, Oct 31 2009

a(n) = A000005(n)^2 - A000005(n^2).
G.f.: Sum_{n>=1} A000005(n^2)*x^(2*n)/(1-x^n). - Mircea Merca, Feb 26 2014
a(n) = A035116(n) - A048691(n). - Amiram Eldar, Apr 25 2024

A069363 Number of 5 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 99, 5503, 247759, 10056959, 384479935, 14142942975, 506544513343, 17792504911231, 615793150236223, 21067276157958271, 714097521397778495, 24022705580163949439, 803089367467759614015, 26706726258154287563903

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Row 5 of A359576.

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A069377 Number of 19 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 22619537, 73209847696773, 115159568055775538305, 127111602733664216603859933, 114600698023505978867449552531361, 90979848541738331379871952593270363301, 66310152669631463041584664319631353072678161, 45499186393097406209322583222635994035907090539853

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

a(5)-a(9) from Sean A. Irvine, Apr 30 2024

A069381 Number of n X 6 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

63, 3367, 167337, 8057905, 384479935, 18287614751, 868972410929, 41278350729313, 1960665141991079, 93127506982471999, 4423369428678533705, 210101996822111263265, 9979493366382754619551, 474010149850018604630031, 22514756623847166766088601

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069382 Number of n X 7 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

127, 14197, 1461797, 144769425, 14142942975, 1374273318721, 133267613878665, 12913642307413193, 1251002814214706415, 121180225103268487525, 11738064445690843712709, 1137001292953873957167377, 110135446775389136829629647, 10668293121854243285020490369

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

cp's OEIS Frontend

A248225 a(n) = 6^n - 3^n.

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A327318 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 1 and s = 1/2.

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A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

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A383755 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 3^(n-k) * T(n-1,k-1) + 4^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

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A065622 Numerator of 1 - (3/4)^n - frac((3/2)^n), where frac(x) = x - floor(x).

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A065814 a(n) = tau(n)^2 - tau(n^2), where tau(n) = A000005(n).

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A069363 Number of 5 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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A069377 Number of 19 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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A069381 Number of n X 6 binary arrays with a path of adjacent 1's from top row to bottom row.

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