A099621 -id:A099621 - cp's OEIS frontend

A249997 Expansion of 1/((1-x)(1+3x)(1-4x)).

1, 2, 15, 40, 221, 702, 3355, 11780, 52041, 193402, 817895, 3138720, 12953461, 50618102, 206059635, 813476860, 3286192481, 13047914802, 52482224575, 209057202200, 838843897101, 3347530323502, 13413657088715, 53584020970740, 214547906035321, 857556157684202

Offset: 0

Alex Ratushnyak, Dec 28 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,11,-12).

Cf. A016208, A099621, A249998, A249999.

[((-1)^n*3^(n+3) +4^(n+3) -7)/84: n in [0..50]]; // G. C. Greubel, Jul 21 2022

LinearRecurrence[{2,11,-12}, {1,2,15}, 50] (* G. C. Greubel, Jul 21 2022 *)

[((-1)^n*3^(n+3) +4^(n+3) -7)/84 for n in (0..50)] # G. C. Greubel, Jul 21 2022

G.f.: 1/((1-x) * (1+3*x) * (1-4*x)).
a(n) = (-1)^n*3^(n+2)/28 + 4^(n+2)/21 -1/12. - R. J. Mathar, Jan 09 2015
E.g.f.: (1/84)*(27*exp(-3*x) - 7*exp(x) + 64*exp(4*x)). - G. C. Greubel, Jul 21 2022

A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)4^(n-k-1)(5/4)^k.

Original entry on oeis.org

0, 1, 8, 53, 316, 1785, 9744, 51997, 273092, 1417889, 7299160, 37334661, 190028748, 963565513, 4871514656, 24572321645, 123720601684, 622038982257, 3123938806632, 15674669614549, 78593250398300, 393845861293721

Offset: 0

Paul Barry, Oct 25 2004

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1) * u^(n-k-1) * (v/u)^(k-1) has g.f. x^2/((1-u*x) * (1-u*x-v*x^2)) and satisfies the recurrence a(n) = 2*u*a(n-1) - (u^2 - v)*a(n-2) - u*v*a(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-11,-20).

Cf. A094705, A099582, A099621.

[(5^(n+2) -6*4^(n+1) -(-1)^n)/30: n in [0..40]]; // G. C. Greubel, Jul 22 2022

LinearRecurrence[{8,-11,-20},{0,1,8},30] (* Harvey P. Dale, Nov 05 2017 *)

[(5^(n+2) -6*4^(n+1) -(-1)^n)/30 for n in (0..40)] # G. C. Greubel, Jul 22 2022

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.
a(n) = 8*a(n-1) - 11*a(n-2) - 20*a(n-3).
G.f.: x^2/((1-4*x)*(1-4*x-5*x^2)) = x^2/((1+x)*(1-4*x)*(1-5*x)).
From G. C. Greubel, Jul 22 2022: (Start)
a(n) = (1/30)*(5^(n+2) - 6*4^(n+1) - (-1)^n).
E.g.f.: (1/30)*(25*exp(5*x) - 24*exp(4*x) - exp(-x)). (End)

A249998 Expansion of 1/((1+x)(1+3x)(1-4x)).

Original entry on oeis.org

1, 0, 13, 12, 169, 312, 2341, 6084, 34177, 107184, 517309, 1803516, 8011225, 29653416, 125788117, 481629108, 1991086513, 7770635808, 31663673965, 124911303660, 504875391241, 2003811035160, 8062315730053, 32108048151972, 128855836912609, 514152414736272, 2060422457687581

Offset: 0

Alex Ratushnyak, Dec 28 2014

Index entries for linear recurrences with constant coefficients, signature (0,13,12).

Cf. A016208, A099621, A249997.

a(n) = (-1)^n*3^(n+2)/14 + 4^(n+2)/35 - (-1)^n/10. - R. J. Mathar, Jan 09 2015

cp's OEIS Frontend

A249997 Expansion of 1/((1-x)(1+3x)(1-4x)).

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A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)4^(n-k-1)(5/4)^k.

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A249998 Expansion of 1/((1+x)(1+3x)(1-4x)).

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cp's OEIS Frontend

A249997 Expansion of 1/((1-x)*(1+3*x)*(1-4*x)).

Views

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Magma

Mathematica

SageMath

Formula

A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.

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Magma

Mathematica

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A249998 Expansion of 1/((1+x)*(1+3*x)*(1-4*x)).

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A249997 Expansion of 1/((1-x)(1+3x)(1-4x)).

A099622 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)4^(n-k-1)(5/4)^k.

A249998 Expansion of 1/((1+x)(1+3x)(1-4x)).