A131130 -id:A131130 - cp's OEIS frontend

A154117 Expansion of (1 - x + 3x^2)/((1-x)(1-2*x)).

1, 2, 7, 17, 37, 77, 157, 317, 637, 1277, 2557, 5117, 10237, 20477, 40957, 81917, 163837, 327677, 655357, 1310717, 2621437, 5242877, 10485757, 20971517, 41943037, 83886077, 167772157, 335544317, 671088637, 1342177277, 2684354557

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

Binomial transform of 1,1,4,1,4,1,4,1,4,1,4,1,4,1,4,... - Philippe Deleham, Jan 05 2009

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

Cf. A094373, A000079, A083329, A095121, A131128, A131130.

[1] cat [5*2^n-3 : n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

Join[{1}, Table[ 5*2^(n - 1) - 3, {n, 1, 10}]] (* or *) Join[{1, 2, 7}, LinearRecurrence[{3, -2}, {17, 37}, 10]] (* G. C. Greubel, Sep 02 2016 *)

a(n)=if(n, 5<<(n-1)-3, 1) \\ Charles R Greathouse IV, Sep 02 2016

From Philippe Deléham, Jan 05 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), n > 2.
a(n) = 2*a(n-1) + 3, n > 1.
a(n) = 5*2^(n-1) - 3, n >= 1. (End)
E.g.f.: (1/2)*(3 - 6*exp(x) + 5*exp(2*x)). - G. C. Greubel, Sep 02 2016

a(0) added by Philippe Deléham, Jan 05 2009

A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

Original entry on oeis.org

1, 2, 11, 29, 65, 137, 281, 569, 1145, 2297, 4601, 9209, 18425, 36857, 73721, 147449, 294905, 589817, 1179641, 2359289, 4718585, 9437177, 18874361, 37748729, 75497465, 150994937, 301989881, 603979769, 1207959545, 2415919097

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,8,1,8,1,8,1,8,1,8,1,8,1,8,...

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

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130

Join[{1},LinearRecurrence[{3,-2},{2,11}, 25]] (* or *) Join[{1},Table[9*2^(n-1) - 7, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+7*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=11.
a(n) = 9*2^(n-1) - 7, n>0, with a(0)=1.
a(n) = 2*a(n-1) + 7, n>1, with a(0)=1, a(1)=2.
From G. C. Greubel, Sep 08 2016: (Start)
a(n) = 9*2^(n-1) - 7 for n >= 1.
E.g.f.: (1/2)*(9*exp(2*x) - 14*exp(x) + 7). (End)

A288732 a(n) = a(n-1) + 2a(n-4) - 2a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 18, 22, 26, 34, 42, 50, 58, 74, 90, 106, 122, 154, 186, 218, 250, 314, 378, 442, 506, 634, 762, 890, 1018, 1274, 1530, 1786, 2042, 2554, 3066, 3578, 4090, 5114, 6138, 7162, 8186, 10234, 12282, 14330, 16378, 20474, 24570, 28666, 32762

Offset: 0

Clark Kimberling, Jun 16 2017

Conjecture:  a(n) is the number of letters (0's and 1's) in the n-th iterate of the mapping 00->1000, 10->01, starting with 00; see A288729.
From Michel Dekking, Mar 25 2018: (Start)
Note that a(n) - a(n-1) = 2*[a(n-4) - a(n-5)] for n>4.
It follows that this sequence is a union of four simple sequences:
a(4k-4) = 4*2^k - 6 = A131130(k) for k = 1,2,3,...
a(4k-3) = 5*2^k - 6 = A020714(k) - 6 for k = 1,2,3...
a(4k-2) = 6*2^k - 6 = A007283(k+1) - 6 for k = 1,2,3, ...
a(4k-1) = 7*2^k - 6 = A048489(k) for k = 1,2,3...
(End)

Clark Kimberling, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2).

Cf. A288729.

a:=[2,4,6,8,10];; for n in [6..45] do a[n]:=a[n-1]+2*a[n-4]-2*a[n-5]; od; a; # Muniru A Asiru, Mar 22 2018

f:= gfun:-rectoproc({a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5),
a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 25 2018

LinearRecurrence[{1, 0, 0, 2, -2}, {2, 4, 8, 8, 10}, 40]

x='x+O('x^99); Vec(2*(1+x+x^2+x^3-x^4)/(1-x-2*x^4+2*x^5)) \\ Altug Alkan, Mar 22 2018

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.

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

a(41)-a(49) from Muniru A Asiru, Mar 22 2018

A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

Original entry on oeis.org

1, 2, 12, 32, 72, 152, 312, 632, 1272, 2552, 5112, 10232, 20472, 40952, 81912, 163832, 327672, 655352, 1310712, 2621432, 5242872, 10485752, 20971512, 41943032, 83886072, 167772152, 335544312, 671088632, 1342177272, 2684354552, 5368709112, 10737418232

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,9,1,9,1,9,1,9,1,9,1,9,1,9,...

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

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130, A154251

Join[{1},LinearRecurrence[{3,-2},{2,12},40]]  (* Harvey P. Dale, Dec 30 2014 *)
Join[{1},Table[5*2^n - 8, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+8*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=12.
a(n) = 2*a(n-1) + 8, n>1, with a(0)=1, a(1)=2.
a(n) = 10*2^(n-1) - 8, n>=1, with a(0)=1.
E.g.f.: 5*exp(2*x) - 8*exp(x) + 4. - G. C. Greubel, Sep 08 2016

Two terms corrected by Johannes W. Meijer, May 26 2011

A177227 Triangle, read by rows, T(n, k) = -binomial(n,k) for 0 < k < n, otherwise T(n, k) = 2.

Original entry on oeis.org

2, 2, 2, 2, -2, 2, 2, -3, -3, 2, 2, -4, -6, -4, 2, 2, -5, -10, -10, -5, 2, 2, -6, -15, -20, -15, -6, 2, 2, -7, -21, -35, -35, -21, -7, 2, 2, -8, -28, -56, -70, -56, -28, -8, 2, 2, -9, -36, -84, -126, -126, -84, -36, -9, 2, 2, -10, -45, -120, -210, -252, -210, -120, -45, -10, 2

Offset: 0

Roger L. Bagula, May 05 2010

This triangle may also be constructed in the following way. Let f_{n}(t) = d^n/dt^n t/(1+t) = (-1)^(n+1)*n!*(1+t)^(-n-1). Then the triangle is given as f_{n}(t)/((1+t)*f_{k}(t)*f_{n-k}(t)) when t = 1/2 (this sequence), t = 1/3 (A177228), and t = 1/4 (A177229).

			Triangle begins as:
  2;
  2,   2;
  2,  -2,   2;
  2,  -3,  -3,    2;
  2,  -4,  -6,   -4,    2;
  2,  -5, -10,  -10,   -5,    2;
  2,  -6, -15,  -20,  -15,   -6,    2;
  2,  -7, -21,  -35,  -35,  -21,   -7,    2;
  2,  -8, -28,  -56,  -70,  -56,  -28,   -8,   2;
  2,  -9, -36,  -84, -126, -126,  -84,  -36,  -9,   2;
  2, -10, -45, -120, -210, -252, -210, -120, -45, -10,  2;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A131130 (related to row sums), A177228, A177229.

A177227:= func< n,k | k eq 0 or k eq n select 2 else -Binomial(n,k) >;
[A177227(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2024

T[n_, k_]:= If[k==0 || k==n, 2, -Binomial[n,k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A177227(n,k): return 2 if (k==0 or k==n) else -binomial(n,k)
flatten([[A177227(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 09 2024

T(n, 0) = T(n, n) = 2, otherwise T(n, k) = -binomial(n,k).
Sum_{k=0..n} T(n, k) = -A131130(n-2) - 3*[n=0], n >= 1 (row sums).
From G. C. Greubel, Apr 09 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = 3*(1 + (-1)^n) - 4*[n=0].
Sum_{k=0..floor(n/2)} T(n-k,k) = (3/2)*(3 + (-1)^n - 2*[n=0])-Fibonacci(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 3*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)

Edited by G. C. Greubel, Apr 09 2024

A246168 a(n) = 2^n - 10.

Original entry on oeis.org

-9, -8, -6, -2, 6, 22, 54, 118, 246, 502, 1014, 2038, 4086, 8182, 16374, 32758, 65526, 131062, 262134, 524278, 1048566, 2097142, 4194294, 8388598, 16777206, 33554422, 67108854, 134217718, 268435446, 536870902, 1073741814, 2147483638

Offset: 0

Vincenzo Librandi, Aug 18 2014

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

Sequences of the form 2^n-k: A000079 (k=0), A000225 (k=1), A000918 (k=2), A036563 (k=3), A028399 (k=4), A168616 (k=5), A131130 (k=6), A048490 (k=7), A159741 (k=8), A185346 (k=9), this sequence (k=10).

Magma
```
[2^n-10: n in [0..40]];
```

Table[2^n - 10, {n, 0, 35}] (* or *) CoefficientList[Series[(-9 + 19 x)/(1 - 3 x + 2 x^2), {x, 0, 35}], x]
LinearRecurrence[{3,-2},{-9,-8},50] (* Harvey P. Dale, Jan 11 2024 *)

vector(50, n, 2^(n-1)-10) \\ Derek Orr, Aug 18 2014

G.f.: (-9+19*x)/(1-3*x+2*x^2).
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A000079(n) - 10.
From Elmo R. Oliveira, Dec 21 2023: (Start)
a(n) = 2*a(n-1) + 10 for n>0.
E.g.f.: exp(x)*(exp(x) - 10). (End)

A131131 4A007318 - 3A097806.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 4, 12, 9, 1, 4, 16, 24, 13, 1, 4, 20, 40, 40, 17, 1, 4, 24, 60, 80, 60, 21, 1, 4, 28, 84, 140, 140, 84, 25, 1, 4, 32, 112, 224, 280, 224, 112, 29, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A131130, (1, 2, 10, 26, 52, 98, 190, ...), the binomial transform of (1, 1, 7, 1, 7, 1, ...). Generally, triangles generated from N*A007318 - (N-1)*A097806 have row sums that are binomial transforms of (1, 1, (N-1), 1, (N-1), 1, ...). A095121 = (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...) and = row sums of A131108.

Triangle T(n,k), 0 <= k <= n,read by rows given by [1,3,-4,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle:
  1;
  1,  1;
  4,  5,  1;
  4, 12,  9,  1;
  4, 16, 24, 13,  1
  4, 20, 40, 40, 17,  1;
  ...

Cf. A131130, A131129, A131128, A131127, A046055, A131108, A095121, A097806.

4*A007318 - 3*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

G.f.: (1-x*y+3*x^2+3*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A185647 Expansion of (1+2x)(1+2x^2)/((1-x)(1+x)(1-2*x^2)).

Original entry on oeis.org

1, 2, 5, 10, 13, 26, 29, 58, 61, 122, 125, 250, 253, 506, 509, 1018, 1021, 2042, 2045, 4090, 4093, 8186, 8189, 16378, 16381, 32762, 32765, 65530, 65533, 131066, 131069, 262138, 262141, 524282, 524285, 1048570, 1048573, 2097146, 2097149, 4194298, 4194301

Offset: 0

Philippe Deléham, Apr 23 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A036563, A131130

LinearRecurrence[{0, 3, 0, -2}, {1, 2, 5, 10}, 50] (* G. C. Greubel, Jul 09 2017 *)

x='x+O('x^50); Vec((1+2*x)*(1+2*x^2)/((1-x)*(1+x)*(1-2*x^2))) \\ G. C. Greubel, Jul 09 2017

a(n) = a(n-1)*2 if n odd.
a(n) = a(n-1)+3 if n even.
a(2n) = 2^(n+2)-3 = A036563(n+2).
a(2n+1) = 2^(n+3)-6 = A131130(n+1).
a(n) = 3*a(n-2) - 2*a(n-4) with a(0)=1, a(1)=2, a(2)=5, a(3)=10.

A306352 a(n) is the least k >= 0 such that all the positive divisors of n have a distinct value under the mapping d -> d AND k (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 2, 7, 10, 13, 2, 15, 4, 5, 6, 15, 16, 31, 2, 29, 6, 7, 2, 31, 12, 9, 10, 11, 4, 15, 2, 31, 42, 49, 6, 63, 4, 7, 6, 63, 8, 15, 2, 14, 14, 5, 2, 63, 18, 29, 18, 21, 4, 31, 6, 23, 18, 9, 2, 31, 4, 5, 14, 63, 76, 127, 2, 115, 6, 15, 2, 127, 8, 13

Offset: 1

Rémy Sigrist, Feb 09 2019

This sequence has similarities with A167234.

Will every nonnegative integer appear in the sequence?

			For n = 15:
- the divisors of 15 are: 1, 3, 5 and 15,
- their values under the mapping d -> d AND k for k = 0..6 are:
  k\d|  1  3  5  15
  ---+-------------
    0|  0  0  0  0
    1|  1  1  1  1
    2|  0  2  0  2
    3|  1  3  1  3
    4|  0  0  4  4
    5|  1  1  5  5
    6|  0  2  4  6
- the first row with 4 distinct values corresponds to k = 6,
- hence a(15) = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n = 1..2^19 (where the color is function of floor(n / 2^A070939(a(n)))).

Cf. A000120, A002145, A028399, A070939, A131130, A167234, A218388.

a(n) = my (d=divisors(n)); for (m=0, oo, if (#Set(apply(v -> bitand(v, m), d))==#d, return (m)))

a(2^k) = 2^k - 1 for any k >= 0.
a(n) = 2 iff n belongs to A002145.
a(n) <= A218388(n).
a(n) AND A218388(n) = a(n).
A000120(a(n)) = 1 iff n is a prime number.
Apparently:
- a(3^k) belongs to A131130 for any k > 0,
- a(5^k) belongs to A028399 for any k >= 0.

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 0, 7, 9, 0, 0, 0, 0, 15, 17, 0, 0, 0, 0, 0, 31, 33, 0, 0, 0, 0, 0, 0, 63, 65, 0, 0, 0, 0, 0, 0, 0, 127, 129, 0, 0, 0, 0, 0, 0, 0, 0, 255, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 511, 513, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047

Offset: 0

Philippe Deléham, Jan 07 2009

Column sums give A003945.

			Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 0, 7, 9;
0, 0, 0, 0, 15, 17; ...

Cf. A000225, A094373

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A040000(n), A094373(n), A000079(n), A083329(n), A095121(n), A154117(n), A131128(n), A154118(n), A131130(n), A154251(n), A154252(n) for x = -1,0,1,2,3,4,5,6,7,8,9 respectively.
G.f.: (1-x*y+x^2*y-x^2*y^2)/(1-3*x*y+2*x^2*y^2). - Philippe Deléham, Nov 02 2013
T(n,k) = 3*T(n-1,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(2,0) = 0, T(2,1) = 1, T(2,2) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 02 2013

cp's OEIS Frontend

A154117 Expansion of (1 - x + 3*x^2)/((1-x)*(1-2*x)).

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A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

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A288732 a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.

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A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

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A177227 Triangle, read by rows, T(n, k) = -binomial(n,k) for 0 < k < n, otherwise T(n, k) = 2.

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A246168 a(n) = 2^n - 10.

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A131131 4*A007318 - 3*A097806.

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A154117 Expansion of (1 - x + 3x^2)/((1-x)(1-2*x)).

A288732 a(n) = a(n-1) + 2a(n-4) - 2a(n-5) for n >= 5, where a(0) = 2, a(1) = 4, a(2) = 6, a(3) = 8, a(4) = 10.

A131131 4A007318 - 3A097806.

A185647 Expansion of (1+2x)(1+2x^2)/((1-x)(1+x)(1-2*x^2)).