A264236 -id:A264236 - cp's OEIS frontend

A264237 Sum of values of vertices at level n of the hyperbolic Pascal pyramid.

1, 3, 9, 33, 165, 1137, 9837, 95193, 962541, 9884889, 102049197, 1055383929, 10921055661, 113032307769, 1169952636525, 12109971475065, 125349031354029, 1297477519769145, 13430093334225645, 139013932289379321, 1438923355509080877, 14894194022848480185

Offset: 0

Michel Marcus, Nov 09 2015

Colin Barker, Table of n, a(n) for n = 0..987
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (6th line of Table 2).
Index entries for linear recurrences with constant coefficients, signature (18,-99,226,-224,92,-12).

Cf. A035344, A264236.

CoefficientList[Series[-(20*x^5 - 8*x^4 + 58*x^3 - 54*x^2 + 15*x - 1)/((x - 1)*(2*x^2 - 4*x + 1)*(6*x^3 - 28*x^2 + 13*x - 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

Vec(-(20*x^5-8*x^4+58*x^3-54*x^2+15*x-1)/((x-1)*(2*x^2-4*x+1)*(6*x^3-28*x^2+13*x-1)) + O(x^30)) \\ Colin Barker, Nov 09 2015

a(n) = 18*a(n-1) - 99*a(n-2) + 226*a(n-3) - 224*a(n-4) + 92*a(n-5) - 12*a(n-6), for n >= 7.

G.f.: -(20*x^5-8*x^4+58*x^3-54*x^2+15*x-1) / ((x-1)*(2*x^2-4*x+1)*(6*x^3-28*x^2+13*x-1)). - Colin Barker, Nov 09 2015

Definition edited by Eric M. Schmidt, Sep 17 2017

A292290 Number of vertices of type A at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 3, 6, 12, 27, 66, 168, 435, 1134, 2964, 7755, 20298, 53136, 139107, 364182, 953436, 2496123, 6534930, 17108664, 44791059, 117264510, 307002468, 803742891, 2104226202, 5508935712, 14422580931, 37758807078, 98853840300, 258802713819, 677554301154

Offset: 0

Eric M. Schmidt, Sep 13 2017

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (1st line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A264236.

CoefficientList[Series[3*x^2*(1 - 2*x)/((1 - x)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(2), Vec(3*x^2*(1 - 2*x) / ((1 - x)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 4.
From Colin Barker, Sep 17 2017: (Start)
G.f.: 3*x^2*(1 - 2*x) / ((1 - x)*(1 - 3*x + x^2)).
a(n) = 3*(1 + (2^(-1-n)*((7-3*sqrt(5))*(3+sqrt(5))^n - (3-sqrt(5))^n*(7+3*sqrt(5)))) / sqrt(5)) for n>0.
(End)
a(n) = 3*(Fibonacci(2*n - 4) + 1) for n > 0. - Ehren Metcalfe, Apr 18 2019

A292291 Number of vertices of type B at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 3, 12, 36, 99, 264, 696, 1827, 4788, 12540, 32835, 85968, 225072, 589251, 1542684, 4038804, 10573731, 27682392, 72473448, 189737955, 496740420, 1300483308, 3404709507, 8913645216, 23336226144, 61095033219, 159948873516, 418751587332, 1096305888483

Offset: 0

Eric M. Schmidt, Sep 13 2017

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (2nd line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A264236.

CoefficientList[Series[3*x^3/((1 - x)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)
LinearRecurrence[{4,-4,1},{0,0,0,3},40] (* Harvey P. Dale, Oct 25 2017 *)

concat(vector(3), Vec(3*x^3 / ((1 - x)*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Sep 17 2017

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 4.
G.f.: 3*x^3 / ((1 - x)*(1 - 3*x + x^2)). - Colin Barker, Sep 17 2017
a(n) = 3*Fibonacci(2*n - 3) - 3 for n > 0. - Ehren Metcalfe, Apr 18 2019

A292292 Number of vertices of type C at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 34, 174, 1128, 8251, 63315, 494175, 3879370, 30512736, 240149088, 1890487729, 14883249459, 117174190329, 922506823618, 7262871367566, 57180440473320, 450180590519275, 3544264121625315, 27903931958216271, 219687190433359498

Offset: 0

Eric M. Schmidt, Sep 13 2017

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (3rd line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (12,-37,37,-12,1).

Cf. A264236.

CoefficientList[Series[x^3*(1 - 9*x + 10*x^2)/((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(3), Vec(x^3*(1 - 9*x + 10*x^2) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5), n >= 6.
G.f.: x^3*(1 - 9*x + 10*x^2) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)). - Colin Barker, Sep 17 2017
a(n) = A001091(n-3)/15 + 3*A002878(n-3)/5 + 1/3 for n > 0. - Ehren Metcalfe, Apr 18 2019

A292293 Number of vertices of type D at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 0, 3, 24, 177, 1347, 10467, 82029, 644808, 5073915, 39939900, 314427960, 2475438408, 19488960504, 153435934587, 1207997701872, 9510543548457, 74876345104299, 589500202673403, 4641125238026805, 36539501601385200, 287674887310843395, 2264859596198883588

Offset: 0

Eric M. Schmidt, Sep 13 2017

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (4th line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (12,-37,37,-12,1).

Cf. A264236.

CoefficientList[Series[3*x^4*(1 - 4*x)/((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(4), Vec(3*x^4*(1 - 4*x) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5), n >= 6.

G.f.: 3*x^4*(1 - 4*x) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)). - Colin Barker, Sep 17 2017

A292294 Number of vertices of type E at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 0, 3, 39, 357, 2952, 23622, 186984, 1474773, 11617815, 91485075, 720308160, 5671099008, 44648794944, 351520074867, 2767513935927, 21788596994037, 171541276628904, 1350541654293318, 10632792057873480, 83711795070905925, 659061569195852295

Offset: 0

Eric M. Schmidt, Sep 13 2017

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (5th line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (12,-37,37,-12,1).

Cf. A264236.

CoefficientList[Series[3*x^4*(1 + x)/((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)
LinearRecurrence[{12,-37,37,-12,1},{0,0,0,0,3,39},30] (* Harvey P. Dale, Oct 09 2018 *)

concat(vector(4), Vec(3*x^4*(1 + x) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5), n >= 6.
G.f.: 3*x^4*(1 + x) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)). - Colin Barker, Sep 17 2017
a(n) = 1 + (A001091(n-2) - 3*Lucas(2*(2-n)))/5 for n > 0. - Ehren Metcalfe, Apr 18 2019

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A264237 Sum of values of vertices at level n of the hyperbolic Pascal pyramid.

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A292290 Number of vertices of type A at level n of the hyperbolic Pascal pyramid.

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A292291 Number of vertices of type B at level n of the hyperbolic Pascal pyramid.

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A292292 Number of vertices of type C at level n of the hyperbolic Pascal pyramid.

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A292293 Number of vertices of type D at level n of the hyperbolic Pascal pyramid.

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A292294 Number of vertices of type E at level n of the hyperbolic Pascal pyramid.

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