A137358 -id:A137358 - cp's OEIS frontend

A137356 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k).

1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 92, 149, 250, 431, 750, 1299, 2227, 3784, 6401, 10828, 18364, 31236, 53228, 90741, 154603, 263178, 447702, 761403, 1295022, 2203162, 3749001, 6380241, 10858285, 18478155, 31443013, 53501860, 91034937, 154900529, 263576791

Offset: 0

Don Knuth, Apr 11 2008

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Robert Israel, Table of n, a(n) for n = 0..4329
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,3).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1).

Cf. A137357-A137361, A136444, A137402.

I:=[1,1,1,1,1]; [n le 5 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, Aug 09 2015

f:= gfun:-rectoproc({a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5),seq(a(i)=1,i=0..4)},a(n),remember):
map(f, [$0..50]); # Robert Israel, May 26 2017

LinearRecurrence[{3, -3, 1, 0, 1}, {1, 1, 1, 1, 1}, 50] (* Vincenzo Librandi, Aug 09 2015 *)
CoefficientList[Series[(1-x)^2/(1 - 3 x + 3 x^2 - x^3 - x^5), {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 09 2015 *)
Table[Sum[Binomial[n-2k,3k],{k,0,n/2}],{n,0,50}] (* Harvey P. Dale, Nov 07 2021 *)

Let A_n = Sum_{k<=n/2}binomial(n-2k,3k) (the present sequence), B_n= Sum_{k<=n/2}binomial(n-2k, 3k+1)(A137357), C_n= Sum_{k<=n/2}binomial(n-2k, 3k+2) (A137358).
Then A_n = A_{n-1} + C_{n-3} + \delta_{n0}, B_n=B_{n-1} + A_{n-1}, C_n=C_{n-1} + B_{n-1};
so the generating functions are A = (1-z)^2/p(z), B=z(1-z)/p(z), C=z^2/p(z),
where p(z) = (1-z)^3 - z^5 = 1 - 3z + 3z^2 - z^3 - z^5.
The growth ratio is the real root of r^2(r-1)^3 = 1, approximately 1.70161.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5). - Vincenzo Librandi, Aug 09 2015
a(n) = hypergeom([-(1/5)*n, -(1/5)*n+1/5, 2/5-(1/5)*n, 3/5-(1/5)*n, -(1/5)*n+4/5], [1/3, 2/3, -(1/2)*n, -(1/2)*n+1/2], -3125/108). - Robert Israel, May 26 2017
G.f.: -(x-1)^2/(-1+3*x-3*x^2+x^3+x^5) . - R. J. Mathar, May 29 2017

A137357 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 12, 23, 44, 80, 138, 230, 379, 629, 1060, 1810, 3109, 5336, 9120, 15521, 26349, 44713, 75949, 129177, 219918, 374521, 637699, 1085401, 1846804, 3141826, 5344988, 9093989, 15474230, 26332515, 44810670, 76253683, 129755543, 220790480

Offset: 0

Don Knuth, Apr 11 2008

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1).

Cf. A137356, A137358, A137359, A137360, A137361, A136444.

LinearRecurrence[{3, -3, 1, 0, 1}, Range[0, 4], 50] (* Paolo Xausa, Mar 17 2024 *)

G.f.: x*(x-1) / (x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A137402 a(n) = Sum_{k=0..n} binomial(floor(n-2k/3), k).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 16, 28, 48, 81, 136, 229, 388, 661, 1129, 1928, 3287, 5594, 9510, 16164, 27484, 46757, 79577, 135454, 230552, 392355, 667620, 1135924, 1932721, 3288563, 5595805, 9522067, 16203273, 27572144, 46917243, 79834375, 135845607, 231154212

Offset: 0

Don Knuth, Apr 11 2008

A_n + B_{n-1} + C_{n-2} in the notation of A137356.

Lim_{n->infinity} a(n+1)/a(n) = x ~= 1.7016..., with x given by the real root (A324498) of (x - 1)^3*x^2 = 1. - Hugo Pfoertner, Mar 15 2019

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1).

Cf. A137356, A137357, A137358, A324498.

[(&+[Binomial(Floor(n-2*k/3), k): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Mar 15 2019

f:=n->add( binomial( floor(n-2*k/3), k), k=0..n);

Table[Sum[Binomial[Floor[n-(2k)/3],k],{k,0,n}],{n,0,40}] (* or *) LinearRecurrence[{3,-3,1,0,1},{1,1,2,3,5},40] (* Harvey P. Dale, Aug 22 2011 *)

Vec((1-2*x+2*x^2-x^3+x^4)/(1-3*x+3*x^2-x^3-x^5) + O(x^50)) \\ Colin Barker, Dec 14 2015

a(n) = sum(k=0, n, binomial(floor(n-2*k/3), k)); \\ Altug Alkan, Dec 14 2015

[sum(binomial(floor(n-2*k/3),k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Mar 15 2019

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5); a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Aug 22 2011

G.f.: (1 - 2*x + 2*x^2 - x^3 + x^4) / (1 - 3*x + 3*x^2 - x^3 - x^5). - Colin Barker, Dec 14 2015

A324498 Decimal expansion of the real solution to x^2*(x-1)^3 = 1.

Original entry on oeis.org

1, 7, 0, 1, 6, 0, 6, 8, 8, 7, 1, 8, 1, 1, 7, 0, 8, 3, 3, 6, 9, 2, 2, 1, 6, 4, 6, 0, 8, 5, 5, 4, 2, 8, 8, 2, 2, 6, 9, 4, 6, 4, 7, 5, 4, 4, 9, 2, 8, 5, 7, 3, 5, 8, 4, 8, 5, 7, 7, 8, 2, 4, 2, 6, 3, 7, 6, 0, 3, 4, 5, 9, 3, 5, 9, 0, 7, 9, 1, 5, 8, 8, 7, 1, 0, 7, 0, 4, 8, 7

Offset: 1

Hugo Pfoertner, Mar 04 2019

Asymptotic ratio of consecutive terms in A137402 for n -> infinity.

			1.701606887181170833692216460855428822694647544928573584857782426376...

Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, p. 207, answer to exercise 126, Addison-Wesley, 2009.

Cf. A137356, A137357, A137358, A137402.

RealDigits[Root[x^2 (x-1)^3-1,1],10,120][[1]] (* Harvey P. Dale, Nov 07 2022 *)

PARI
```
solve(x=1,2,x^2*(x-1)^3-1)
```

A138777 Triangle read by rows: T(n,k)=binomial(n-2k,3k+2) (n>=2, 0<=k<=(n-2)/5).

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 1, 28, 6, 36, 21, 45, 56, 55, 126, 66, 252, 1, 78, 462, 9, 91, 792, 45, 105, 1287, 165, 120, 2002, 495, 136, 3003, 1287, 1, 153, 4368, 3003, 12, 171, 6188, 6435, 78, 190, 8568, 12870, 364

Offset: 2

Emeric Deutsch, May 10 2008

Row n contains floor((n+3)/5) terms.

Row sums yield A137358.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Cf. A137358.

T:=proc(n,k) options operator, arrow: binomial(n-2*k,3*k+2) end proc: for n from 2 to 20 do seq(T(n,k),k=0..(n-2)*1/5) end do; # yields sequence in triangular form

Flatten[Table[Binomial[n-2k,3k+2],{n,2,20},{k,0,(n-2)/5}]] (* Harvey P. Dale, Mar 07 2013 *)

cp's OEIS Frontend

A137356 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k).

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A137357 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+1).

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A137402 a(n) = Sum_{k=0..n} binomial(floor(n-2k/3), k).

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A324498 Decimal expansion of the real solution to x^2*(x-1)^3 = 1.

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A138777 Triangle read by rows: T(n,k)=binomial(n-2k,3k+2) (n>=2, 0<=k<=(n-2)/5).

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