A387507 -id:A387507 - cp's OEIS frontend

A387508 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(n-3*k,k)^2.

1, 1, 1, 1, 3, 9, 19, 33, 55, 109, 243, 529, 1071, 2093, 4179, 8673, 18255, 37981, 77923, 159649, 329935, 687117, 1432403, 2977505, 6179215, 12841597, 26757059, 55840033, 116551119, 243209325, 507658803, 1060551137, 2217515151, 4639042909, 9707403811

Offset: 0

Seiichi Manyama, Aug 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001850, A108488, A387507.

Cf. A246883, A375293.

[(&+[2^k * Binomial(n-3*k, k)^2: k in [0..Floor(n/4)]]): n in [0..40]]; // Vincenzo Librandi, Sep 02 2025

Table[Sum[2^k*Binomial[n-3*k, k]^2,{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)

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

G.f.: 1/sqrt((1-x-2*x^4)^2 - 8*x^5).

A387510 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 2, 4, 10, 32, 104, 324, 1000, 3136, 9992, 32064, 103168, 332816, 1077152, 3497024, 11381920, 37121280, 121285760, 396922944, 1300906112, 4269367296, 14028169344, 46143475712, 151932559360, 500710965504, 1651533562368, 5451595506688, 18008220715520

Offset: 0

Seiichi Manyama, Aug 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A387507, A387509.

Cf. A246840, A387513.

[(&+[2^(n-2*k) * Binomial(n-2*k, k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[2^(n-2*k)*Binomial[n-2*k,k]^2,{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, Sep 01 2025 *)

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

G.f.: 1/sqrt((1-2*x-2*x^3)^2 - 16*x^4).

A387600 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^k * binomial(2n-4k+2,2*k+1).

Original entry on oeis.org

1, 2, 3, 8, 25, 62, 139, 340, 877, 2186, 5311, 13056, 32497, 80566, 198595, 490092, 1212597, 2999522, 7411207, 18311384, 45265657, 111902478, 276579275, 683566148, 1689580733, 4176276794, 10322491919, 25513684656, 63061990401, 155871106406, 385266605779

Offset: 0

Seiichi Manyama, Sep 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (2,-1,4,4,0,-4).

Cf. A387507.

[&+[2^k* Binomial(2*n-4*k+2, 2*k+1)/2: k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 03 2025

Table[Sum[2^k*Binomial[2*n-4*k+2, 2*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 03 2025 *)

a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k+2, 2*k+1))/2;

G.f.: B(x)^2, where B(x) is the g.f. of A387507.
G.f.: 1/((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).

cp's OEIS Frontend

A387508 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(n-3*k,k)^2.

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A387510 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(n-2*k,k)^2.

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Magma

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Formula

A387600 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^k * binomial(2n-4k+2,2*k+1).

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cp's OEIS Frontend

A387508 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(n-3*k,k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387510 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(n-2*k,k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387600 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k+2,2*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387510 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(n-2*k,k)^2.

A387600 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^k * binomial(2n-4k+2,2*k+1).