A107293 -id:A107293 - cp's OEIS frontend

A173693 a(n) = ceiling(A107293(n)/2).

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 5, 7, 10, 14, 20, 28, 41, 59, 85, 122, 176, 254, 367, 529, 764, 1102, 1591, 2296, 3313, 4782, 6901, 9960, 14375, 20747, 29944, 43217, 62373, 90021, 129925, 187516, 270636, 390601, 563742, 813631, 1174288, 1694813, 2446070

Offset: 0

Roger L. Bagula, Nov 25 2010

Cf. A107293.

A107293 := proc(n)
    option remember;
    if n <=4 then
        op(n+1,[0,0,0,0,1]) ;
    else
        procname(n-1)+procname(n-2)-procname(n-3)+procname(n-5) ;
    end if;
end proc:
A173693 := proc(n)
    ceil(A107293(n)/2) ;
end proc: # R. J. Mathar, Feb 18 2016

M = {{0, 1, 0, 0, 0},
{0, 0, 1, 0, 0},
{0, 0, 0, 1, 0},
{0, 0, 0, 0, 1},
{1, 0, -1, 1, 1}}
v[0] = {0, 0, 0, 0, 1}
v[n_] := v[n] = M.v[n - 1]
Table[v[n][[1]] - Floor[v[n][[1]]/2], {n, 0, 30}]

a(n) = A107293(n) - floor(A107293(n)/2) = ceiling(A107293(n)/2).

Conjecture: a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-5) + a(n-31) - a(n-32) - a(n-33) + a(n-34) - a(n-36). - R. J. Mathar, Feb 18 2016

A185357 Expansion of 1/(1 - x - x^2 + x^18 - x^20).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4180, 6763, 10942, 17703, 28642, 46340, 74974, 121301, 196254, 317521, 513720, 831152, 1344728, 2175647, 3519998, 5695035, 9214046, 14907484, 24118947, 39022252, 63134437, 102145749

Offset: 0

Roger L. Bagula, Jan 21 2012

Limiting ratio is 1.61791..., the real root of -1 + x^2 - x^18 - x^19 + x^20. Signature in Mathematica is:
-CoefficientList[1 - x - x^2 + x^18 - x^20, x]
{-1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1}.
The sequence agrees with the Fibonacci numbers (A000045) for the first 18 terms.

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Terr and Eric W. Weisstein, MathWorld: Pisot Number
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1).

Cf. A117791, A107293, A204631.

CoefficientList[Series[1/(1 - x - x^2 + x^18 - x^20), {x, 0, 50}], x]

Vec(1/(1-x-x^2+x^18-x^20) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

A202907 Expand 1/(1 - (3/2)x + (2/3)x^4 - x^5) in powers of x, then multiply coefficient of x^n by 3^floor(n/4)*2^n to get integers.

Original entry on oeis.org

1, 3, 9, 27, 211, 633, 1899, 5697, 52297, 156891, 470673, 1412019, 12675403, 38026209, 114078627, 342235881, 3081171505, 9243514515, 27730543545, 83191630635, 748691121283, 2246073363849, 6738220091547

Offset: 0

Roger L. Bagula, Jan 27 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,211,0,0,0,7776).

Cf. A167602, A167602, A117791, A107293, A204631, A185357.

Table[3^Floor[k/4]*2^k*SeriesCoefficient[ Series[1/(1 - (3/2)* x + (2/3) x^4 - x^5), {x, 0, 30}], k], {k, 0, 30}] (* Bagula *)
a[ n_] := 2^n 3^Quotient[ n, 4] SeriesCoefficient[ 1 / (1 - 3/2 x + 2/3 x^4 - x^5), {x, 0, n}] (* Michael Somos, Jan 27 2012 *)

From Chai Wah Wu, Aug 01 2020: (Start)
a(n) = 211*a(n-4) + 7776*a(n-8) for n > 7.
G.f.: -(3*x + 1)*(9*x^2 + 1)/(7776*x^8 + 211*x^4 - 1). (End)

A225484 Expansion of 1/(1 - x^3 - x^4 - x^5 - x^6 + x^9).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 6, 8, 10, 14, 20, 26, 36, 49, 66, 90, 123, 167, 227, 308, 420, 571, 776, 1056, 1436, 1952, 2656, 3612, 4912, 6680, 9085, 12356, 16804, 22853, 31081, 42269, 57486, 78182, 106327, 144604

Offset: 0

Roger L. Bagula, May 08 2013

Limiting ratio is 1.3599997117115008..., the largest real root of 1 - x^3 - x^4 - x^5 - x^6 + x^9 = 0.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,1,0,0,-1).

Cf. A117791, A107293, A204631, A225391, A225482.

SeriesCoefficient[Series[1/(1 - x^3 - x^4 - x^5 - x^6 + x^9), {x, 0, 50}], n]

Vec(1/(1-x^3-x^4-x^5-x^6+x^9)+O(x^99)) \\ Charles R Greathouse IV, May 08 2013

A124445 Expansion of 1/(1-x-xy+x^2y-x^3*y^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 5, 1, 1, 1, 6, 9, 13, 12, 6, 1, 1, 1, 7, 11, 19, 22, 17, 7, 1, 1, 1, 8, 13, 26, 35, 35, 23, 8, 1, 1, 1, 9, 15, 34, 51, 61, 53, 30, 9, 1

Offset: 0

Paul Barry, Nov 01 2006

Row sums are A107293(n+4). Diagonal sums are A005314(n+1).

Reversal of A124279. T(2n,n) is A002426. - Paul Barry, Nov 01 2006

			Triangle begins:
1,
1, 1,
1, 1, 1,
1, 1, 2, 1,
1, 1, 3, 3, 1,
1, 1, 4, 5, 4, 1,
1, 1, 5, 7, 8, 5, 1,
1, 1, 6, 9, 13, 12, 6, 1,
1, 1, 7, 11, 19, 22, 17, 7, 1

Cf. A180562.

Number triangle T(n,k)=sum{j=0..n, C(j,k-j)*C(n-k,k-j)}*[k<=n];

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + T(n-3,k-2), T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2014

A205961 Expansion of 1/(-32x^5 + 8x^3 - 4*x^2 - x + 1).

Original entry on oeis.org

1, 1, 5, 1, 13, 9, 85, 177, 477, 921, 1701, 4289, 9389, 28201, 60917, 153041, 308349, 733625, 1645125, 4062177, 9670989, 22625865, 52288405, 118067953, 276204317, 639640537, 1523941861

Offset: 0

Roger L. Bagula, Feb 02 2012

Previous name was: Expand 1/(1 - x/2 - x^2 + x^3 - x^5) in powers of x, then multiply coefficient of x^n by 2^n to get integers.
The sequence is from -1 + x^2 - x^3 - x^4/2 + x^5 with real root 1.1647612555333289.
The limiting ratio of successive terms is 2*1.1647612555333289.
Recurrence: -32 *a (n) + 8 *a (n + 2) - 4 *a (n + 4) + a (n + 5) == 0; with a (1) == 1; a (2) == 1; a (3) == 5; a (4) == 1; a (5) == 13 (from FindSequenceFunction[]).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-8,0,32).

Cf. A202907, A167602, A167602, A117791, A107293, A204631, A185357.

CoefficientList[Series[1/(1 - x/2 - x^2 + x^3 - x^5), {x, 0, 50}], x] * 2^Range[0, 50]
LinearRecurrence[{1,4,-8,0,32}, {1,1,5,1,13}, 100] (* G. C. Greubel, Nov 16 2016 *)

for(n=0,30, print1(2^n*polcoeff(1/(1-x/2 - x^2 + x^3 - x^5) + O(x^32), n), ", ")) \\ G. C. Greubel, Nov 16 2016

New name from Joerg Arndt, Nov 19 2016

A206568 Expand 1/(8 - 8 x + 3 x^3 - 2 x^4) in powers of x, then multiply coefficient of x^n by 8^(1 + floor(n/3)) to get integers.

Original entry on oeis.org

1, 1, 1, 5, 4, 3, 25, 23, 22, 149, 130, 110, 785, 693, 623, 4389, 3880, 3397, 23977, 21115, 18684, 131893, 116502, 102680, 724705, 638985, 563949, 3980357, 3512812, 3098935, 21873593, 19295871, 17024690

Offset: 0

Roger L. Bagula, Feb 09 2012

Bob Hanlon (hanlonr(AT)cox.net) helped convert the expansion to a recursion.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A202907, A167602, A167602, A117791, A107293, A204631, A185357, A205961.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <64|69|21|-1>>^ iquo(n, 3, 'r'). `if`(r=0, <<1, 5, 25, 149>>, `if`(r=1, <<1, 4, 23, 130>>, <<1, 3, 22, 110>>)))[1, 1]: seq (a(n), n=0..40); # Alois P. Heinz, Feb 11 2012

(* expansion*)
Table[8^(1 + Floor[n/3])*SeriesCoefficient[Series[1/(8 - 8 x + 3 x^3 - 2 x^4), {x, 0, 50}], n], {n, 0,50}]
(*recursion*)
a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 5; a[5] = 4; a[6] = 3;
a[7] = 25; a[8] = 23; a[9] = 22; a[10] = 149; a[11] = 130;
a[12] = 110;
a[n_Integer?Positive] := a[n] = 64*a[-12 + n] + 69*a[-9 + n] + 21*a[-6 +n] - a[-3 + n]
Table[a[n], {n, 1, 50}]

G.f.: (-4*x^8-6*x^7-9*x^6-4*x^5-5*x^4-6*x^3-x^2-x-1) / (64*x^12 +69*x^9 +21*x^6 -x^3-1).

A225490 Expansion of 1/(1 - x - x^2 + x^5 + x^6 - x^7).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 30, 42, 60, 84, 118, 166, 233, 327, 458, 643, 901, 1263, 1770, 2481, 3477, 4872, 6828, 9568, 13408, 18788, 26328, 36893, 51697, 72442, 101511, 142245, 199323, 279306, 391383, 548433

Offset: 0

Roger L. Bagula, May 08 2013

Limiting ratio is 1.401268367939855..., the largest real root of -1 + x + x^2 - x^5 - x^6 + x^7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, -1, 1).

Cf. A117791, A107293, A204631, A225391, A225482, A225484.

CoefficientList[Series[1/(1 - x - x^2 + x^5 + x^6 - x^7), {x, 0, 50}], x]

A225500 Expansion of 1/(1 - x - x^5 + x^6 - x^7 - x^11 + x^12).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 33, 43, 56, 73, 95, 124, 161, 210, 273, 355, 463, 603, 786, 1023, 1332, 1735, 2259, 2942, 3831, 4989, 6497, 8461, 11019, 14350, 18687, 24335, 31691, 41270, 53745

Offset: 0

Roger L. Bagula, May 09 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1,1,0,0,0,1,-1).

Cf. A117791, A107293, A204631, A225391, A225482, A225484, A225490, A225490.

CoefficientList[Series[1/(1 - x - x^5 + x^6 - x^7 - x^11 + x^12), {x, 0, 50}], x]
LinearRecurrence[{1,0,0,0,1,-1,1,0,0,0,1,-1},{1,1,1,1,1,2,2,3,4,5,7,9},100] (* G. C. Greubel, Nov 16 2016 *)

Vec(1/(1-x -x^5 +x^6 -x^7 -x^11 +x^12) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

A107332 Expansion of x^2*(-1+x+x^2)/(-1+x+x^2-x^3+x^5).

Original entry on oeis.org

0, 1, 0, 0, -1, -1, -1, -1, -1, -2, -3, -5, -7, -10, -14, -20, -29, -42, -61, -88, -127, -183, -264, -381, -550, -794, -1146, -1654, -2387, -3445, -4972, -7176, -10357, -14948, -21574, -31137, -44939, -64859, -93609, -135103, -194990, -281423, -406169, -586211, -846060, -1221092, -1762364

Offset: 1

Roger L. Bagula, Jun 08 2005

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1).

a[0]:=0:a[1]:=1:a[2]:=0:a[3]:=0:a[4]:=-1:
for n from 5 to 46 do a[n]:=a[n-1]+a[n-2]-a[n-3]+a[n-5] od:
seq(a[n],n=0..46);

LinearRecurrence[{1,1,-1,0,1},{0,1,0,0,-1},50] (* Harvey P. Dale, Oct 11 2015 *)

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-5) for n>=5.
O.g.f.: x^2*(-1+x+x^2)/(-1+x+x^2-x^3+x^5). - R. J. Mathar, Dec 02 2007
a(n) = A107293(n+2)-A107293(n+1)-A107293(n). - R. J. Mathar, Dec 17 2017

Edited by N. J. A. Sloane, May 13 2006

New name using g.f. from Joerg Arndt, Dec 26 2022

cp's OEIS Frontend

A173693 a(n) = ceiling(A107293(n)/2).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A185357 Expansion of 1/(1 - x - x^2 + x^18 - x^20).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A202907 Expand 1/(1 - (3/2)*x + (2/3)*x^4 - x^5) in powers of x, then multiply coefficient of x^n by 3^floor(n/4)*2^n to get integers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A225484 Expansion of 1/(1 - x^3 - x^4 - x^5 - x^6 + x^9).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A124445 Expansion of 1/(1-x-x*y+x^2*y-x^3*y^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A205961 Expansion of 1/(-32*x^5 + 8*x^3 - 4*x^2 - x + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A206568 Expand 1/(8 - 8 x + 3 x^3 - 2 x^4) in powers of x, then multiply coefficient of x^n by 8^(1 + floor(n/3)) to get integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A225490 Expansion of 1/(1 - x - x^2 + x^5 + x^6 - x^7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A225500 Expansion of 1/(1 - x - x^5 + x^6 - x^7 - x^11 + x^12).

Views

A202907 Expand 1/(1 - (3/2)x + (2/3)x^4 - x^5) in powers of x, then multiply coefficient of x^n by 3^floor(n/4)*2^n to get integers.

A124445 Expansion of 1/(1-x-xy+x^2y-x^3*y^2).

A205961 Expansion of 1/(-32x^5 + 8x^3 - 4*x^2 - x + 1).