A027082 -id:A027082 - cp's OEIS frontend

A027107 a(n) = Sum_{k=0..2n} (k+1) * A027082(n, 2n-k).

1, 6, 20, 62, 188, 566, 1700, 5102, 15308, 45926, 137780, 413342, 1240028, 3720086, 11160260, 33480782, 100442348, 301327046, 903981140, 2711943422, 8135830268, 24407490806, 73222472420, 219667417262, 659002251788

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

[1] cat [7*3^n-1: n in [0..30]]; // Vincenzo Librandi, Mar 24 2013

Join[{1}, Table[7 3^(n-1) - 1, {n, 30}]] (* Vincenzo Librandi, Mar 24 2013 *)

For n>0, a(n) = 7*3^(n-1) - 1.
G.f.: (1+2*x-x^2)/(1-4*x+3*x^2). [Bruno Berselli, Mar 25 2013]
a(n) = 2*A237930(n-1), n>0. - R. J. Mathar, Jun 24 2020

A027083 a(n) = A027082(n, n+2).

Original entry on oeis.org

2, 6, 14, 28, 54, 102, 190, 352, 650, 1198, 2206, 4060, 7470, 13742, 25278, 46496, 85522, 157302, 289326, 532156, 978790, 1800278, 3311230, 6090304, 11201818, 20603358, 37895486, 69700668, 128199518, 235795678, 433695870

Offset: 2

Clark Kimberling

Paolo Xausa, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Cf. A008937, A027024, A027082.

LinearRecurrence[{2, 0, 0, -1}, {2, 6, 14, 28}, 50] (* Paolo Xausa, Sep 16 2024 *)

G.f.: (2x^2(1+x+x^2))/((1-x)(1-x-x^2-x^3)). Cf. A008937.
a(n) = A027024(n) + 1.
a(n) = A000213(n+3) -3. - R. J. Mathar, Jun 24 2020

A027086 a(n) = A027082(n, n+4).

Original entry on oeis.org

11, 41, 108, 246, 517, 1035, 2010, 3828, 7199, 13429, 24920, 46090, 85065, 156791, 288758, 531528, 978099, 1799521, 3310404, 6089406, 11200845, 20602307, 37894354, 69699452, 128198215, 235794285, 433694384, 797689490, 1467180945, 2698567791, 4963441390

Offset: 4

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-1,2,-1).

Cf. A027026, A027082.

I:=[11,41,108,246,517,1035]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3)-Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Feb 20 2016

LinearRecurrence[{4, -5, 2, -1, 2, -1}, {11, 41, 108, 246, 517, 1035}, 35] (* Vincenzo Librandi, Feb 20 2016 *)

Vec(x^4*(11-3*x-x^2-3*x^3+2*x^4)/((1-x)^3*(1-x-x^2-x^3)) + O(x^40)) \\ Colin Barker, Feb 20 2016

a(n) = A027026(n) + (n+1)(n+2)/2 - 3.
From Colin Barker, Feb 20 2016: (Start)
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3)-a(n-4)+2*a(n-5)-a(n-6) for n>9.
G.f.: x^4*(11-3*x-x^2-3*x^3+2*x^4) / ((1-x)^3*(1-x-x^2-x^3)).
(End)
a(n) = A000213(n+4) -4 -3*n*(n+3)/2. - R. J. Mathar, Jun 24 2020

A027098 T(n,n) + T(n,n+1) + ... + T(n,2n), T given by A027082.

Original entry on oeis.org

1, 2, 6, 19, 55, 161, 469, 1362, 3954, 11483, 33379, 97146, 283114, 826214, 2414366, 7064217, 20693797, 60686847, 178152055, 523473326, 1539479414, 4531059224, 13345835288, 39335816471, 116012701523, 342356498884

Offset: 0

Clark Kimberling

nonn

A027085 a(n) = A027082(n, n+3).

Original entry on oeis.org

5, 16, 39, 84, 169, 328, 623, 1168, 2173, 4024, 7431, 13700, 25233, 46448, 85471, 157248, 289269, 532096, 978727, 1800212, 3311161, 6090232, 11201743, 20603280, 37895405, 69700584, 128199431, 235795588, 433695777, 797690976

Offset: 0

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A027025, A027082.

CoefficientList[Series[-((-x^3+x^2+x+5)/((x-1)^2 (x^3+x^2+x-1))),{x,0,30}],x] (* Harvey P. Dale, May 15 2013 *)

G.f.: (x^3-x^2-x-5)/((x-1)^2(x^3+x^2+x-1)). - Harvey P. Dale, May 15 2013
a(n) = A027025(n) + n + 1.
a(n) = A000213(n+6) -3*(n+4). - R. J. Mathar, Jun 24 2020

A027087 a(n) = A027082(n, 2n).

Original entry on oeis.org

1, 1, 2, 5, 11, 27, 68, 175, 458, 1215, 3258, 8815, 24031, 65937, 181936, 504473, 1404879, 3927495, 11017802, 31004871, 87497297, 247559933, 702089528, 1995483853, 5682959332, 16214622573, 46343364290, 132667706857, 380358924209

Offset: 0

Clark Kimberling

nonn

Cf. A027082.

Conjecture: D-finite with recurrence +3*n*a(n) +3*(-5*n+3)*a(n-1) +(7*n-4)*a(n-2) +(25*n-67)*a(n-3) +(17*n-44)*a(n-4) +(19*n-97)*a(n-5) +(5*n-32)*a(n-6) +3*(n-7)*a(n-7)=0. - R. J. Mathar, Jun 24 2020

A027088 a(n) = A027082(n, 2n-1).

Original entry on oeis.org

1, 3, 6, 16, 41, 107, 283, 757, 2043, 5557, 15216, 41906, 115999, 322537, 900406, 2522616, 7090307, 19987069, 56492426, 160062636, 454529595, 1293394325, 3687475479, 10531663241, 30128741717, 86324342567, 247691217352

Offset: 1

Clark Kimberling

nonn

Cf. A027082.

Conjecture: D-finite with recurrence 3*(n+1)*(18*n-85)*a(n) +(292*n^2-1316*n-255)*a(n-1) +(-218*n^2+1213*n-1056)*a(n-2) +2*(-248*n^2+1666*n-2223)*a(n-3) +(-50*n^2-497*n+3243)*a(n-4) +(-52*n^2+80*n+765)*a(n-5) +3*(22*n-83)*(n-6)*a(n-6)=0. - R. J. Mathar, Jun 24 2020

A027089 a(n) = A027082(n, 2n-2).

Original entry on oeis.org

1, 5, 14, 39, 108, 299, 828, 2299, 6401, 17875, 50062, 140601, 395933, 1117737, 3162812, 8969267, 25487555, 72565339, 206969662, 591304797, 1691991626, 4848703909, 13914119144, 39980978277, 115023510495, 331303274449

Offset: 2

Clark Kimberling

nonn

Cf. A027082.

Conjecture: D-finite with recurrence -(n+2)*(244*n-1065)*a(n) +3*(n+1)*(422*n -1703)*a(n-1) +2*(-359*n^2 +1211*n -1420)*a(n-2) +2*(-1174*n^2 +6319*n -7410)*a(n-3) +(-760*n^2 +2173*n +1616)*a(n-4) +(-502*n^2 +2245*n -1123)*a(n-5) +6*(23*n-67) *(n-5)*a(n-6)=0. - R. J. Mathar, Jun 24 2020

A027090 a(n) = A027082(n, 2n-3).

Original entry on oeis.org

1, 3, 9, 28, 84, 246, 714, 2059, 5917, 16971, 48633, 139333, 399267, 1144669, 3283839, 9427981, 27090715, 77911897, 224272499, 646157234, 1863317958, 5377939756, 15535195892, 44913790501, 129955421387, 376313609029

Offset: 2

Clark Kimberling

nonn

A027091 a(n) = A027082(n, 2n-4).

Original entry on oeis.org

1, 1, 5, 17, 54, 169, 517, 1559, 4653, 13787, 40638, 119333, 349469, 1021433, 2981330, 8693467, 25333627, 73795263, 214915073, 625855927, 1822630172, 5308552227, 15464475465, 45060652609, 131334677147, 382907083385

Offset: 2

Clark Kimberling

nonn

Cf. A027082.

cp's OEIS Frontend

A027107 a(n) = Sum_{k=0..2n} (k+1) * A027082(n, 2n-k).

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A027083 a(n) = A027082(n, n+2).

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A027086 a(n) = A027082(n, n+4).

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A027098 T(n,n) + T(n,n+1) + ... + T(n,2n), T given by A027082.

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A027085 a(n) = A027082(n, n+3).

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Mathematica

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A027087 a(n) = A027082(n, 2n).

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A027088 a(n) = A027082(n, 2n-1).

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A027089 a(n) = A027082(n, 2n-2).

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A027090 a(n) = A027082(n, 2n-3).

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A027091 a(n) = A027082(n, 2n-4).

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