A387623 -id:A387623 - cp's OEIS frontend

A108480 Expansion of (1-x-2x^2)/(1-2x-3x^2-4x^3+4*x^4).

1, 1, 3, 13, 35, 117, 379, 1197, 3859, 12357, 39563, 126845, 406371, 1302101, 4172443, 13369293, 42838835, 137266917, 439837739, 1409354397, 4515934339, 14470215157, 46366299963, 148569565165, 476055153491, 1525403341701

Offset: 0

Paul Barry, Jun 04 2005

Index entries for linear recurrences with constant coefficients, signature (2,3,4,-4).

Cf. A001541, A387622, A387623.

CoefficientList[Series[(1-x-2*x^2)/(1-2*x-3*x^2-4*x^3+4*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)
LinearRecurrence[{2,3,4,-4},{1,1,3,13},30] (* Harvey P. Dale, Aug 29 2023 *)

a(n) = 2*a(n-1) + 3*a(n-2) + 4*a(n-3) - 4*a(n-4).
a(n) = Sum_{k=0..floor(n/2)} C(2*(n-k), 2k) * 2^k.
a(n) ~ (1+sqrt((4*sqrt(2)-1)/31)) * (1+2*sqrt(2)+sqrt(1+4*sqrt(2)))^n/2^(n+2). - Vaclav Kotesovec, Jul 24 2013

A387622 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2n-4k,2*k).

Original entry on oeis.org

1, 1, 1, 3, 13, 31, 61, 151, 413, 1031, 2445, 5991, 15069, 37447, 91917, 226503, 561373, 1389735, 3431501, 8474983, 20955229, 51814407, 128054029, 316455559, 782209629, 1933537511, 4779082829, 11812031271, 29195752157, 72164132167, 178368130061

Offset: 0

Seiichi Manyama, Sep 03 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,4,4,0,-4).

Cf. A001541, A108480, A387623.

[&+[2^k* Binomial(2*n-4*k, 2*k): k in [0..Floor (n/3)]]: n in [0..30]]; // Vincenzo Librandi, Sep 05 2025

Table[Sum[2^k*Binomial[2*n-4*k,2*k],{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 05 2025 *)

a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k, 2*k));

G.f.: (1-x-2*x^3)/((1-x-2*x^3)^2 - 8*x^4).

a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3) + 4*a(n-4) - 4*a(n-6).

A387629 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2n-6k+1,2*k+1).

Original entry on oeis.org

1, 3, 5, 7, 11, 31, 83, 183, 351, 675, 1435, 3231, 7119, 14987, 30963, 64871, 138775, 298403, 636091, 1344191, 2838399, 6021371, 12818467, 27277207, 57911207, 122790675, 260485131, 553185519, 1175285967, 2496108459, 5298760307, 11246985927, 23877452663, 50702334403

Offset: 0

Seiichi Manyama, Sep 03 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,4,4,0,0,-4).

Cf. A001653, A387627, A387628.

Cf. A387623, A387626.

[&+[2^k* Binomial(2*n-6*k+1, 2*k+1): k in [0..Floor (n/4)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Table[Sum[2^k*Binomial[2*n-6*k+1,2*k+1],{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *)

a(n) = sum(k=0, n\4, 2^k*binomial(2*n-6*k+1, 2*k+1));

G.f.: (1+x-2*x^4)/((1+x-2*x^4)^2 - 4*x).

a(n) = 2*a(n-1) - a(n-2) + 4*a(n-4) + 4*a(n-5) - 4*a(n-8).

cp's OEIS Frontend

A108480 Expansion of (1-x-2x^2)/(1-2x-3x^2-4x^3+4*x^4).

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A387622 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2n-4k,2*k).

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A387629 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2n-6k+1,2*k+1).

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cp's OEIS Frontend

A108480 Expansion of (1-x-2*x^2)/(1-2*x-3*x^2-4*x^3+4*x^4).

Views

Author

Keywords

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Crossrefs

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Mathematica

Formula

A387622 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k,2*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387629 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2*n-6*k+1,2*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A108480 Expansion of (1-x-2x^2)/(1-2x-3x^2-4x^3+4*x^4).

A387622 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2n-4k,2*k).

A387629 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2n-6k+1,2*k+1).