A014825 a(n) = 4*a(n-1) + n with n > 1, a(1)=1.

1, 6, 27, 112, 453, 1818, 7279, 29124, 116505, 466030, 1864131, 7456536, 29826157, 119304642, 477218583, 1908874348, 7635497409, 30541989654, 122167958635, 488671834560, 1954687338261, 7818749353066

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Hacène Belbachir and El-Mehdi Mehiri, Enumerating moves in the optimal solution of the Tower of Hanoi, arXiv:2210.08657 [math.CO], 2022.

László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), 461-469.

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Formula

a(n) = (4^(n+1) - 3*n - 4)/9.

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

a(n) = Sum_{k=0..n} (n-k)*4^k = Sum_{k=0..n} k*4^(n-k). - Paul Barry, Jul 30 2004

a(n) = Sum_{k=0..n} binomial(n+2, k+2)*3^k [Offset 0]. - Paul Barry, Jul 30 2004

a(n) = Sum_{k=0..n} Sum_{j=0..2k} (-1)^(j+1)*J(j)*J(2k-j), J(n) = A001045(n). - Paul Barry, Oct 23 2009

Convolution square of A006314. - Michael Somos, Jun 20 2012

E.g.f.: (4*exp(4*x) - (4+3*x)*exp(x))/9. - G. C. Greubel, Feb 18 2020

a(n) = A014916(-n-1)*4^(n+1) = A091919(2*n-2) for all n in Z. - Michael Somos, Oct 02 2020

a(n) = Sum_{k=0..n} A002450(k). - Joseph Brown, May 11 2021

Last digits give A171654. - Paul Curtz, Oct 10 2021

Example

G.f. = x + 6*x^2 + 27*x^3 + 112*x^4 + 453*x^5 + 1818*x^6 + 7279*x^7 + ...

Mathematica

RecurrenceTable[{a[1]==1, a[n]==4a[n-1]+n}, a[n], {n, 30}] (* Harvey P. Dale, Oct 12 2011 *)
a[ n_]:= SeriesCoefficient[x/((1-4x)(1-x)^2), {x, 0, n}] (* Michael Somos, Jun 20 2012 *)

Prog

(Magma) [(4^(n+1)-3*n-4)/9: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011
(PARI) {a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* Michael Somos, Jun 20 2012 */
(Sage) [(4^(n+1) -3*n -4)/9 for n in (1..30)] # G. C. Greubel, Feb 18 2020

Crossrefs

Cf. A002450 (first differences), A052161 (partial sums).

Cf. A001045, A006314, A014916, A053142, A091919.

Cf. A171654 (mod 10).

Sequence in context: A371965 A198694 A220101 * A141844 A176476 A079742

Adjacent sequences: A014822 A014823 A014824 * A014826 A014827 A014828

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved