OFFSET
1,2
COMMENTS
Coefficients in the hypergeometric series identity 1 - 5*x/(x + 4) + 16*x*(x - 1)/((x + 4)*(x + 6)) - 44*x*(x - 1)*(x - 2)/((x + 4)*(x + 6)*(x + 8)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289. - Peter Bala, May 30 2019
For n>=2, a(n) is the total number of ones in runs of ones of length >=5 over all binary strings of length n+3. - Félix Balado, Aug 06 2025
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..500
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Hypercubes and Isometric Words based on Swap and Mismatch Distance, arXiv:2303.09898 [math.CO], 2023.
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Generalized Fibonacci Cubes Based on Swap and Mismatch Distance, From Strings to Graphs, and Back Again, Open Acc. Ser. Informat. (OASIcs, 2025) Art. 5, 5:1-5:14. See p. 7.
Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 111.
Frank K. Hwang and Colin L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70(2) (1995) 323-333.
Index entries for linear recurrences with constant coefficients, signature (4,-4).
FORMULA
G.f.: x*(1+x)/(1-2*x)^2.
a(n) = (3*n-1) * 2^(n-2).
E.g.f.: exp(2*x)*(1+3*x). The sequence 0, 1, 5, 16, ... has a(n) = ((3n-1)*2^n + 0^n)/4 (offset 0). It is the binomial transform of A032766. The sequence 1, 5, 16, ... has a(n) = (2+3n)*2^(n-1) (offset 0). It is the binomial transform of A016777. - Paul Barry, Jul 23 2003
Row sums of A132776(n-1). - Gary W. Adamson, Aug 29 2007
a(n+1) = det(f(i-j+1))_{1 <= i, j <= n}, where f(0) = 1, f(1) = 5 and for k > 0, we have f(k+1) = 9 and f(-k) = 0. - Mircea Merca, Jun 23 2012
MATHEMATICA
ListCorrelate[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* or *) ListConvolve[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* Ross La Haye, Feb 24 2007 *)
LinearRecurrence[{4, -4}, {1, 5}, 35] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Array[(3# - 1) 2^(# - 2) &, 35] (* Alonso del Arte, Sep 04 2018 *)
CoefficientList[Series[(1 + x)/(1 - 2 * x)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 04 2018 *)
PROG
(PARI) a(n)=if(n<1, 0, (3*n-1)*2^(n-2))
(PARI) a(n)=(3*n-1)<<(n-2) \\ Charles R Greathouse IV, Apr 17 2012
(Magma) [(3*n-1)*2^(n-2): n in [1..50]]; // Vincenzo Librandi, May 09 2011
(Haskell)
a053220 n = a056242 (n + 1) n -- Reinhard Zumkeller, May 08 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Asher Auel, Jan 01 2000
STATUS
approved