A081625 -id:A081625 - cp's OEIS frontend

A178590 a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).

1, 3, 4, 9, 7, 12, 13, 27, 16, 21, 19, 36, 25, 39, 40, 81, 43, 48, 37, 63, 40, 57, 55, 108, 61, 75, 64, 117, 79, 120, 121, 243, 124, 129, 91, 144, 85, 111, 100, 189, 103, 120, 97, 171, 112, 165, 163, 324, 169, 183, 136, 225, 139, 192, 181, 351, 196, 237, 199, 360, 241

Offset: 1

Gary W. Adamson, May 29 2010

In groups of 1, 2, 4, 8, ... terms; sums of group terms appears to be A081625: (1, 7, 41, 223,...), for example: 41 = (9 + 7 + 12 + 13).
Equals row 3 in the array shown in A178568, an infinite family of sequences of the form a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).
Let M = an infinite lower triangular matrix with (1, 3, 1, 0, 0, 0,...) in each column, and with successive columns shifted down twice from the previous column. A178590 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence.
The Stern polynomial B(n,x) evaluated at x=3. See A125184. - T. D. Noe, Feb 28 2011

			In groups of 2^n terms (n=0,1,2,...):
1;
3, 4;
9, 7, 12, 13;
27, 16, 21, 19, 36, 25, 39, 40;
...
a(6) = 12 = 3*a(3) = 3*4
a(7) = 13 = a(3) + a(4) = 4 + 9

Antti Karttunen, Table of n, a(n) for n = 1..8191

Row 3 of A178568.

Cf. A081625, A090880, A260443.

a[0] = a[1] = 1; a[n_] := a[n] = If[ OddQ@n, a[(n - 1)/2] + a[(n + 1)/2], 3*a[n/2]]; Array[a, 61] (* Robert G. Wilson v, Jun 11 2010 *)

a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).
a(n) = A090880(A260443(n)). - Antti Karttunen, Jul 29 2015
G.f.: x * Product_{k>=0} (1 + 3*x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Jul 07 2019

a(19) onwards from Robert G. Wilson v, Jun 11 2010

A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528

Offset: 0

Klaus Brockhaus, Dec 09 2009

Binomial transform of A048473; second binomial transform of A151821; third binomial transform of A010684; fourth binomial transform of A084633 without second term 0; fifth binomial transform of A168589.
Inverse binomial transform of A081625; second inverse binomial transform of A081626; third inverse binomial transform of A081627.
Partial sums of A010036.
Essentially first differences of A006095.
a(n) = A109241(n) converted from binary to decimal. - Robert Price, Jan 19 2016
a(n) is the area enclosed by a Hilbert curve with unit length segments after n iterations, when the start and end points are joined. - Jennifer Buckley, Apr 23 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jennifer Buckley, The Area Enclosed by a Hilbert Curve with Unit Length Segments
Gordon Hamilton, Cookie Monster Counts Cookies, Youtube video, 2010.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A006516 (2^(n-1)*(2^n-1)), A020522 (4^n-2^n), A048473 (2*3^n-1), A151821 (powers of 2, omitting 2 itself), A010684 (repeat 1, 3), A084633 (inverse binomial transform of repeated odd numbers), A168589 ((2-3^n)*(-1)^n), A081625 (2*5^n-3^n), A081626 (2*6^n-4^n), A081627 (2*7^n-5^n), A010036 (sum of 2^n, ..., 2^(n+1)-1), A006095 (Gaussian binomial coefficient [n, 2] for q=2), A171472, A171473.

[2*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

LinearRecurrence[{6,-8},{1,6},30] (* Harvey P. Dale, Aug 02 2020 *)

m=23; v=concat([1, 6], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v

a(n) = Sum_{k=1..2^n-1} k.
a(n) = 2*4^n - 2^n.
G.f.: 1/((1-2*x)*(1-4*x)).
a(n) = A006516(n+1).
a(n) = 4*a(n-1) + 2^n for n > 0, a(0)=1. - Vincenzo Librandi, Jul 17 2011
a(n) = Sum_{k=0..n} 2^(n+k). - Bruno Berselli, Aug 07 2013
a(n) = A020522(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: exp(2*x)*(2*exp(2*x) - 1). - Stefano Spezia, Dec 10 2021

A081626 2*6^n-4^n.

Original entry on oeis.org

1, 8, 56, 368, 2336, 14528, 89216, 543488, 3293696, 19893248, 119883776, 721399808, 4336787456, 26054279168, 156459892736, 939296227328, 5637924847616, 33836139020288, 203051193860096, 1218444602114048

Offset: 0

Paul Barry, Mar 25 2003

Binomial transform of A081625. Inverse binomial transform of A081627.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-24).

[2*6^n-4^n: n in [0..25]]; // Vincenzo Librandi, Aug 09 2013

Table[2 6^n - 4^n, {n, 0, 20}] (* or *) LinearRecurrence[{10, -24}, {1, 8}, 20] (* Harvey P. Dale, Aug 19 2012 *)
CoefficientList[Series[(1 - 2 x) / ((1 - 4 x) (1 - 6 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 09 2013 *)

a(n)=2*6^n-4^n \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 10*a(n-1)-24*a(n-2), a(0)=1, a(1)=8.
G.f.: (1-2*x)/((1-4*x)(1-6*x)).
E.g.f. 2*exp(6*x)-exp(4*x).

A125185 Triangle read by rows: T(n,k) is the coefficient of t^k in the polynomial S(n,t)=[(1+t)(2+t)^n+(1-t)t^n]/2 (0<=k<=n).

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 10, 9, 4, 8, 24, 28, 16, 5, 16, 56, 80, 60, 25, 6, 32, 128, 216, 200, 110, 36, 7, 64, 288, 560, 616, 420, 182, 49, 8, 128, 640, 1408, 1792, 1456, 784, 280, 64, 9, 256, 1408, 3456, 4992, 4704, 3024, 1344, 408, 81, 10, 512, 3072, 8320, 13440, 14400

Offset: 0

Emeric Deutsch, Dec 04 2006

Sum of terms in row n = 3^n. The polynomials S(n,t) can be defined recursively by S(0,t)=1, S(n,t)=t^n - t^(n-1) + (2+t)S(n-1,t) for n>=1. S(n,t)=Sum(B(j,t), j=2^n .. 2^(n+1)-1), where B(n,t) are the Stern polynomials, defined by B(0,t)=0, B(1,t)=1, B(2n,t)=tB(n,t), B(2n+1,t)=B(n+1,t)+B(n,t) for n>=1 (see S. Klavzar et al. and A125184). For example, S(2,t)=B(4,t)+B(5,t)+B(6,t)+B(7,t).

Subtriangle of (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

			Triangle starts:
  1;
  1,2;
  2,4,3;
  4,10,9,4;
  8,24,28,16,5;
  16,56,80,60,25,6;
Triangle (0,1,1,0,0,0,...) DELTA (1,1,-1,1,0,0,0,0,...) begins:
  1
  0, 1
  0, 1, 2
  0, 2, 4, 3
  0, 4, 10, 9, 4
  0, 8, 24, 28, 16, 5
  0, 16, 56, 80, 60, 25, 6

S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007), 86-95.

Cf. A125184.

Maple
```
T:=proc(n,k) if k
				
```

T(n,k)=2^(n-k-1)*(n+k+1)binomial(n,k)/(n-k+1) if k

G.f.: (1-x)/((1-y*x)*(1-(y+2)*x)). - Philippe Deléham, Feb 26 2012

Recurrence : T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 26 2012

Sum_{k=0..n} T(n,k)*x^k = A033999(n), A011782(n), A000244(n), A010036(n), A081625(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 26 2012

A200864 Expansion of 1/((1+x)(1-3x)(1-5x)).

Original entry on oeis.org

1, 7, 42, 230, 1211, 6237, 31732, 160300, 806421, 4046867, 20278622, 101525970, 508028431, 2541337897, 12710276712, 63562145240, 317843011241, 1589311911327, 7946850122002, 39735122306110, 198678226618851, 993398978359157, 4967018427590492, 24835162745336580

Offset: 0

Bruno Berselli, Nov 23 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000.
Index entries for linear recurrences with constant coefficients, signature (7,-7,-15).

Cf. A016209, A200859.

m:=24; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1+x)*(1-3*x)*(1-5*x))));

CoefficientList[Series[1/((1+x)(1-3x)(1-5x)), {x, 0, 24}], x]
LinearRecurrence[{7,-7,-15},{1,7,42},30] (* Harvey P. Dale, May 26 2015 *)

makelist(coeff(taylor(1/((1+x)*(1-3*x)*(1-5*x)), x, 0, n), x, n), n, 0, 23);

PARI
```
Vec(1/((1+x)*(1-3*x)*(1-5*x))+O(x^24))
```

G.f.: 1/((1+x)*(1-3*x)*(1-5*x)).
a(n) = (50*5^n-27*3^n+(-1)^n)/24.
a(n) = 2*a(n-1)+3*a(n-2)+5^n for n>1, a(0)=1, a(1)=7.
a(n) = 7*a(n-1)-7*a(n-2)-15*a(n-3) for n>2, a(0)=1, a(1)=7, a(2)=42.
a(n+1)+a(n) = A005059(n+2).
a(n+2)-a(n) = A081625(n+2).

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cp's OEIS Frontend

A178590 a(2n) = 3*a(n), a(2n+1) = a(n) + a(n+1).

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A171476 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1, a(0)=1, a(1)=6.

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A081626 2*6^n-4^n.

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A125185 Triangle read by rows: T(n,k) is the coefficient of t^k in the polynomial S(n,t)=[(1+t)(2+t)^n+(1-t)t^n]/2 (0<=k<=n).

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A200864 Expansion of 1/((1+x)*(1-3*x)*(1-5*x)).

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A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.

A200864 Expansion of 1/((1+x)(1-3x)(1-5x)).