A026244 a(n) = 4^n*(4^n+1)/2.
1, 10, 136, 2080, 32896, 524800, 8390656, 134225920, 2147516416, 34359869440, 549756338176, 8796095119360, 140737496743936, 2251799847239680, 36028797153181696, 576460752840294400, 9223372039002259456, 147573952598266347520, 2361183241469182345216
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- P. J. Szablowski, On moments of Cantor and related distributions, arXiv preprint arXiv:1403.0386 [math.PR], 2014.
- Index entries for linear recurrences with constant coefficients, signature (20,-64)
Crossrefs
Cf. A052539.
Programs
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Magma
[4^n*(4^n+1)/2: n in [0..30]]; // Vincenzo Librandi, May 01 2011
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Maple
seq(binomial(-4^n, 2), n=0..18); # Zerinvary Lajos, Feb 22 2008
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Mathematica
Table[4^n (4^n + 1)/2, {n, 0, 19}] (* Alonso del Arte, Jun 18 2019 *) LinearRecurrence[{20,-64},{1,10},20] (* Harvey P. Dale, Mar 09 2025 *) -
PARI
a(n)=4^n*(4^n+1)/2 \\ Charles R Greathouse IV, Oct 07 2015
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Scala
((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).map((n: BigInt) => n * (n + 1)/2) // Alonso del Arte, Jun 22 2019
Formula
From Paul Barry, Mar 11 2004: (Start)
With interpolated zeros 0, 1, 0, 10, ... has a(n) = (4^n - (-4)^n + 2*2^n - 2*(-2)^n)/16 and counts walks of length n between adjacent vertices of the 4-cube.
G.f.: (1 - 10*x)/((1 - 4*x)*(1 - 16*x)). (End)
From Philippe Deléham, Sep 08 2009: (Start)
a(n) = Sum_{k = 0..n} 9*k*binomial(2n, 2k) = Sum_{k = 0..n} 9^k*A086645(n, k);
a(n) = 8^n*T(n,5/4) where T is the Chebyshev polynomial of first kind;
e.g.f.: exp(10*x)*cosh(6*x). (End)
a(n) = (2*(n+1))! * [x^(2*(n+1))] (cosh(x)^4-1)/4. - Vladimir Kruchinin, Oct 19 2016
a(n) = 64^n * a(-n) for all n in Z. - Michael Somos, Jul 02 2017