A264237 -id:A264237 - cp's OEIS frontend

A264236 Number of vertices at level n of the hyperbolic Pascal pyramid.

1, 3, 6, 13, 36, 138, 736, 4908, 36351, 280228, 2190651, 17206203, 135357481, 1065387963, 8387050686, 66029196613, 519841755036, 4092692363058, 32221664474776, 253680537891828, 1997222414704551, 15724098193422028, 123795561597659331, 974640390569138163

Offset: 0

Michel Marcus, Nov 09 2015

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (6th line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (12,-37,37,-12,1).

Cf. A076765, A027941, A055588, A264237.

LinearRecurrence[{12, -37, 37, -12, 1}, {1, 3, 6, 13, 36}, 30] (* Bruno Berselli, Nov 09 2015 *)

Vec((1-9*x+7*x^2+15*x^3+3*x^4)/((1-x)*(1-3*x+x^2)*(1-8*x+x^2)) + O(x^50)) \\ Altug Alkan, Nov 09 2015

a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5).
a(n) = (-3/2 + 9*sqrt(5)/10)*((3 + sqrt(5))/2)^n + (-3/2 - 9*sqrt(5)/10)*((3 - sqrt(5))/2)^n + (7/12 - 3*sqrt(15)/20)*(4 + sqrt(15))^n + (7/12 + 3*sqrt(15)/20)*(4 - sqrt(15))^n + 17/6. (See Németh paper, page 9.)
G.f.: (1 - 9*x + 7*x^2 + 15*x^3 + 3*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 - 8*x + x^2)). [Bruno Berselli, Nov 09 2015]
a(n) = A076765(n-3) + 3*Fibonacci(2*(n-1)) + 3. - Ehren Metcalfe, Apr 18 2019

More terms from Bruno Berselli, Nov 09 2015

A292295 Sum of values of vertices of type A at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 6, 18, 54, 174, 582, 1974, 6726, 22950, 78342, 267462, 913158, 3117702, 10644486, 36342534, 124081158, 423639558, 1446395910, 4938304518, 16860426246, 57565095942, 196539531270, 671027933190, 2291032670214, 7822074814470, 26706233917446, 91180786040838

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 2).
Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Cf. A264237.

CoefficientList[Series[6*x^2*(1 - 2*x)/((1 - x)*(1 - 4*x + 2*x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(2), Vec(6*x^2*(1 - 2*x) / ((1 - x)*(1 - 4*x + 2*x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

a(n) = 5*a(n-1) - 6*a(n-2) + 2*a(n-3), n >= 4.
From Colin Barker, Sep 17 2017: (Start)
G.f.: 6*x^2*(1 - 2*x) / ((1 - x)*(1 - 4*x + 2*x^2)).
a(n) = (-3/2)*(-4 + (4-3*sqrt(2))*(2+sqrt(2))^n + (2-sqrt(2))^n*(4+3*sqrt(2))) for n>0.
(End)

A292296 Sum of values of vertices of type B at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 6, 30, 114, 402, 1386, 4746, 16218, 55386, 189114, 645690, 2204538, 7526778, 25698042, 87738618, 299558394, 1022756346, 3491908602, 11922121722, 40704669690, 138974435322, 474488401914, 1620004737018, 5531042144250, 18884159102970, 64474552123386

Offset: 0

Eric M. Schmidt, Sep 14 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 2).
Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Cf. A264237.

CoefficientList[Series[6*x^3/((1 - x)*(1 - 4*x + 2*x^2)), {x, 0, 30}],
x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(3), Vec(6*x^3 / ((1 - x)*(1 - 4*x + 2*x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

a(n) = 5*a(n-1) - 6*a(n-2) + 2*a(n-3), n >= 4.
From Colin Barker, Sep 17 2017: (Start)
G.f.: 6*x^3 / ((1 - x)*(1 - 4*x + 2*x^2)).
a(n) = (1/2)*(-12 + (9-6*sqrt(2))*(2+sqrt(2))^n + (2-sqrt(2))^n*(9+6*sqrt(2))) for n>0.
(End)

A292297 Sum of values of vertices of type C at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 6, 36, 210, 1452, 12138, 114684, 1147002, 11729148, 120902202, 1249686492, 12929303130, 133809210108, 1384977143610, 14335551770268, 148385432561562, 1535924231893308, 15898233466089210, 164561459781232092, 1703363953470584922, 17631399812695032444

Offset: 0

Eric M. Schmidt, Sep 14 2017

Colin Barker, Table of n, a(n) for n = 0..988
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (3rd line of Table 2).
Index entries for linear recurrences with constant coefficients, signature (18,-99,226,-224,92,-12).

Cf. A264237.

CoefficientList[Series[6*x^3*(1 - 12*x + 26*x^2 - 20*x^3)/((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(3), Vec(6*x^3*(1 - 12*x + 26*x^2 - 20*x^3) / ((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)) + O(x^30))) \\ Colin Barker, Sep 17 2017

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), n >= 7.

G.f.: 6*x^3*(1 - 12*x + 26*x^2 - 20*x^3) / ((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)). - Colin Barker, Sep 17 2017

A292298 Sum of values of vertices of type D at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 0, 24, 324, 3600, 38148, 398112, 4132596, 42818208, 443356212, 4589665248, 47509091508, 491769434400, 5090291998452, 52689326584800, 545383755284532, 5645229662006688, 58433377222329972, 604839778633231200, 6260653947359090868, 64803587809297981728

Offset: 0

Eric M. Schmidt, Sep 14 2017

Colin Barker, Table of n, a(n) for n = 0..987
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (4th line of Table 2).
Index entries for linear recurrences with constant coefficients, signature (18,-99,226,-224,92,-12).

Cf. A264237.

I:=[0,0,0,0,24,324,3600]; [n le 7 select I[n] else 18*Self(n-1)-99*Self(n-2)+226*Self(n-3)-224*Self(n-4)+ 92*Self(n-5)-12*Self(n-6): n in [1..30]]; // Vincenzo Librandi, Sep 17 2017

Join[{0}, LinearRecurrence[{18, -99, 226, -224, 92, -12}, {0, 0, 0, 24, 324, 3600}, 20] ] (* Vincenzo Librandi, Sep 17 2017 *)
CoefficientList[Series[12*x^4*(2 - 9*x + 12*x^2)/((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

concat(vector(4), Vec(12*x^4*(2 - 9*x + 12*x^2) / ((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)) + O(x^30))) \\ Colin Barker, Sep 17 2017

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), n >= 7.

G.f.: 12*x^4*(2 - 9*x + 12*x^2) / ((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)). - Colin Barker, Sep 17 2017

A292299 Sum of values of vertices of type E at level n of the hyperbolic Pascal pyramid.

Original entry on oeis.org

0, 0, 0, 0, 18, 312, 3798, 41544, 438270, 4566120, 47368110, 490668936, 5080145070, 52588590888, 544355820750, 5634640292424, 58323941179182, 603707608725096, 6248936971173390, 64682313170747016, 669522088312069614, 6930176023749038760, 71733763792342350798

Offset: 0

Eric M. Schmidt, Sep 14 2017

Colin Barker, Table of n, a(n) for n = 0..987
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (5th line of Table 2).
Index entries for linear recurrences with constant coefficients, signature (18,-99,226,-224,92,-12).

Cf. A264237.

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

concat(vector(4), Vec(6*x^4*(3 - 2*x - 6*x^2) / ((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)) + O(x^30))) \\ Colin Barker, Sep 17 2017

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), n >= 7.

G.f.: 6*x^4*(3 - 2*x - 6*x^2) / ((1 - x)*(1 - 4*x + 2*x^2)*(1 - 13*x + 28*x^2 - 6*x^3)). - Colin Barker, Sep 17 2017

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A264236 Number of vertices at level n of the hyperbolic Pascal pyramid.

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A292295 Sum of values of vertices of type A at level n of the hyperbolic Pascal pyramid.

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A292296 Sum of values of vertices of type B at level n of the hyperbolic Pascal pyramid.

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A292297 Sum of values of vertices of type C at level n of the hyperbolic Pascal pyramid.

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A292298 Sum of values of vertices of type D at level n of the hyperbolic Pascal pyramid.

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A292299 Sum of values of vertices of type E at level n of the hyperbolic Pascal pyramid.

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