A171654 -id:A171654 - cp's OEIS frontend

A197916 Related to the periodic sequence A171654.

0, 1, 6, 7, 2, 3, 8, 9, 4, 5, 10, 11, 6, 7, 12, 13, 8, 9, 14, 15, 10, 11, 16, 17, 12, 13, 18, 19, 14, 15, 20, 21, 16, 17, 22, 23, 18, 19, 24, 25, 20, 21, 26, 27, 22, 23, 28, 29, 24, 25, 30, 31, 26, 27, 32, 33, 28, 29, 34, 35, 30, 31, 36, 37, 32, 33, 38, 39

Offset: 0

Philippe Deléham, Oct 19 2011

All numbers appear twice except numbers 0,1,2,3,4,5 which appear once. - From A. Salagnad, Oct 19 2011.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A171654.

LinearRecurrence[{1,0,0,1,-1},{0,1,6,7,2},80] (* Harvey P. Dale, Jul 30 2019 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^n*[0;1;6;7;2])[1,1] \\ Charles R Greathouse IV, Jul 06 2017

a(n)=A171654(n) mod 10.
a(n)=Sum_k>=0 {A030308(n,k)*b(k)} with b(0)=1, b(1)=6 and b(k)=2^(k-1) for k>1.
G.f. ( -x*(-1-5*x-x^2+5*x^3) ) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 20 2011
a(n) = n/2+(11-(-1)^n)/4 -5*A057077(n)/2. - R. J. Mathar, Oct 20 2011

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
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).

Cf. A002450 (first differences), A052161 (partial sums).
Cf. A001045, A006314, A014916, A053142, A091919.
Cf. A171654 (mod 10).

[(4^(n+1)-3*n-4)/9: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

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 *)

{a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* Michael Somos, Jun 20 2012 */

def A014825(n): return (((1<<(n+1<<1))-4)//3-n)//3 # Chai Wah Wu, Nov 12 2024

[(4^(n+1) -3*n -4)/9 for n in (1..30)] # G. C. Greubel, Feb 18 2020

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

cp's OEIS Frontend

A197916 Related to the periodic sequence A171654.

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