A152174 -id:A152174 - cp's OEIS frontend

A147703 Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 13, 27, 20, 7, 1, 34, 80, 73, 35, 9, 1, 89, 234, 252, 151, 54, 11, 1, 233, 677, 837, 597, 269, 77, 13, 1, 610, 1941, 2702, 2225, 1199, 435, 104, 15, 1, 1597, 5523, 8533, 7943, 4956, 2158, 657, 135, 17, 1

Offset: 0

Paul Barry, Nov 10 2008

Equal to A062110*A007318 when A062110 is regarded as a triangle read by rows.

			Triangle begins
   1;
   1,   1;
   2,   3,   1;
   5,   9,   5,   1;
  13,  27,  20,   7,  1;
  34,  80,  73,  35,  9,  1;
  89, 234, 252, 151, 54, 11, 1;

Indranil Ghosh, Rows 0..100, flattened

Row sums are A006012. Diagonal sums are A147704.

Cf. A062110, A007318, A206800, A001519, A321620.

# The function RiordanSquare is defined in A321620:
RiordanSquare(1 / (1 - x / (1 - x / (1 - x))), 10); # Peter Luschny, Jan 26 2020

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 2*x)/(1 - (3 + y)*x + (1 + y)*x^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 11 2017 *)

Riordan array ((1-2x)/(1-3x+x^2), x(1-x)/(1-3x+x^2)).
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Dec 01 2008
G.f.: (1-2*x)/(1-(3+y)*x+(1+y)*x^2). - Philippe Deléham, Nov 26 2011
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), for n > 1. - Philippe Deléham, Feb 12 2012
The Riordan square of the odd indexed Fibonacci numbers A001519. - Peter Luschny, Jan 26 2020

A087206 a(n) = 2a(n-1) + 4a(n-2); with a(0)=1, a(1)=4.

Original entry on oeis.org

1, 4, 12, 40, 128, 416, 1344, 4352, 14080, 45568, 147456, 477184, 1544192, 4997120, 16171008, 52330496, 169345024, 548012032, 1773404160, 5738856448, 18571329536, 60098084864, 194481487872, 629355315200, 2036636581888

Offset: 0

Paul Barry, Aug 25 2003

Binomial transform of A056487. Unsigned version of A152174.
Number of words of length n over the alphabet {1,2,3,4} such that no odd letter is followed by an odd letter. - Armend Shabani, Feb 18 2017
From Sean A. Irvine, Jun 06 2025: (Start)
Also, the number of walks of length n starting at 0 in the following graph:
  1---2
  |\ /|
  | 0 |
  |/ \|
  4---3. (End)

Jens Christian Claussen, Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration, arXiv:math/0410429 [math.CO], 2004. See Table II, p. 4.
Sean A. Irvine, Walks on Graphs.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A060925, A063782, A253064.

Equals (1/2) * A063727(n-1). Cf. A006483.

LinearRecurrence[{2, 4}, {1, 4}, 25] (* Jean-François Alcover, Sep 21 2017 *)

G.f.: (1+2x)/(1-2x-4x^2).
a(n) = (1-sqrt(5))^n*(1/2-3*sqrt(5)/10)+(1+sqrt(5))^n*(1/2+3*sqrt(5)/10).
a(n) = 2^n*Fibonacci(n+2). - Paul Barry, Mar 22 2004
a(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/sqrt(80). Offset 2. a(4)=12. - Al Hakanson (hawkuu(AT)gmail.com), Apr 11 2009
G.f.: 1/(-2x-1/(-2x-1)). - Paul Barry, Mar 24 2010

Comment corrected by Philippe Deléham, Nov 27 2008

A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 8, 12, 9, 4, 1, 0, 16, 28, 25, 14, 5, 1, 0, 32, 64, 66, 44, 20, 6, 1, 0, 64, 144, 168, 129, 70, 27, 7, 1, 0, 128, 320, 416, 360, 225, 104, 35, 8, 1, 0, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 0, 512, 1536, 2400, 2528, 1970

Offset: 0

Henry Bottomley, May 30 2001

The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
From Paul Barry, Nov 10 2008: (Start)
As number triangle, Riordan array (1, x(1-x)/(1-2x)). A062110*A007318 is A147703.
[0,1,1,0,0,0,....] DELTA [1,0,0,0,.....]. (Philippe Deléham's DELTA is defined in A084938.) (End)
Modulo 2, this triangle T becomes triangle A106344. - Philippe Deléham, Dec 18 2008

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1, 0,  0,   0,   0,    0,    0,     0,     0,     0, ...
  1, 1,  2,   4,   8,   16,   32,    64,   128,   256, ...
  1, 2,  5,  12,  28,   64,  144,   320,   704,  1536, ...
  1, 3,  9,  25,  66,  168,  416,  1008,  2400,  5632, ...
  1, 4, 14,  44, 129,  360,  968,  2528,  6448, 16128, ...
  1, 5, 20,  70, 225,  681, 1970,  5500, 14920, 39520, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, ...
  ... - _Petros Hadjicostas_, Feb 15 2021
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   4,   5,   3,   1;
  0,   8,  12,   9,   4,   1;
  0,  16,  28,  25,  14,   5,   1;
  0,  32,  64,  66,  44,  20,   6,   1;
  0,  64, 144, 168, 129,  70,  27,   7,   1;
  0, 128, 320, 416, 360, 225, 104,  35,   8,   1;
  ... - _Philippe Deléham_, Nov 30 2008

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq., 21 (2018), #18.1.4.

Rows of A include A000007, A011782, A045623, A058396, A062109, A169792, A169793, A169794, A169795, A169796, A169797.
Columns of A include A000012, A001477, A000096, A000297.
Main diagonal of A is A002002.
Table A(n, k) is a multiple of 2^(k-n); dividing by this gives a table similar to A050143 except at the edges.
Essentially the same array as A105306, A160232.

t[n_, n_] = 1; t[n_, k_] := 2^(n-2*k)*k*Hypergeometric2F1[1-k, n-k+1, 2, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Philippe Deléham + symbolic sum *)

a(i,j)=if(i<0 || j<0,0,polcoeff(((1-x)/(1-2*x)+x*O(x^j))^i,j))

Formulas for the square array (A(n,k): n,k >= 0):
A(n, k) = A(n-1, k) + Sum_{0 <= j < k} A(n, j) for n >= 1 and k >= 0 with A(0, k) = 0^k for k >= 0.
G.f.: 1/(1-x*(1-y)/(1-2*y)) = Sum_{i, j >= 0} A(i, j) x^i*y^j.
From Petros Hadjicostas, Feb 15 2021: (Start)
A(n,k) = 2^(k-n)*n*hypergeom([1-n, k+1], [2], -1) for n >= 0 and k >= 1.
A(n,k) = 2*A(n,k-1) + A(n-1,k) - A(n-1,k-1) for n,k >= 1 with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1. (End)
Formulas for the triangle (T(n,k): 0 <= k <= n):
From Philippe Deléham, Aug 01 2006: (Start)
T(n,k) = A121462(n+1,k+1)*2^(n-2*k) for 0 <= k < n.
T(n,k) = 2^(n-2*k)*k*hypergeom([1-k, n-k+1], [2], -1) for 0 <= k < n. (End)
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Dec 09 2008
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 1 <= k <= n-1 with T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013
G.f.: Sum_{n.k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/(x^2*y - x*y - 2*x + 1). - Petros Hadjicostas, Feb 15 2021

Various sections edited by Petros Hadjicostas, Feb 15 2021

A152185 a(n) = -3a(n-1) + 5a(n-2), n > 1; a(0)=1, a(1)=-5.

Original entry on oeis.org

1, -5, 20, -85, 355, -1490, 6245, -26185, 109780, -460265, 1929695, -8090410, 33919705, -142211165, 596232020, -2499751885, 10480415755, -43940006690, 184222098845, -772366329985, 3238209484180, -13576460102465

Offset: 0

Philippe Deléham, Nov 28 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,5).

Cf. A152163, A152166, A152167, A152174.

LinearRecurrence[{-3, 5}, {1, -5}, 25] (* Paolo Xausa, Jan 19 2024 *)

Vec((1-2*x)/(1+3*x-5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

G.f.: (1-2x)/(1+3x-5x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*(-6)^k.
a(n) = (-1)^n*A152187(n). - Philippe Deléham, Nov 29 2008

A099843 A transform of the Fibonacci numbers.

Original entry on oeis.org

1, -5, 21, -89, 377, -1597, 6765, -28657, 121393, -514229, 2178309, -9227465, 39088169, -165580141, 701408733, -2971215073, 12586269025, -53316291173, 225851433717, -956722026041, 4052739537881, -17167680177565, 72723460248141, -308061521170129, 1304969544928657

Offset: 0

Paul Barry, Oct 27 2004

The g.f. is the transform of the g.f. of A000045 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1 + 2(k+1)*x - (2*k-1)*x^2).

Pisano period lengths: 1, 1, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20, ... - R. J. Mathar, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (-4,1).

Cf. A000045, A001076, A015448, A099842.

Cf. A084326 (shifted unsigned inverse binomial transform), A152174 (binomial transform).

[(-1)^n*Fibonacci(3*n+2): n in [0..40]]; // G. C. Greubel, Apr 20 2023

a:= n-> (<<0|1>, <1|-4>>^n.<<1, -5>>)[1,1]:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 21 2023

CoefficientList[Series[(1-x)/(1+4*x-x^2), {x,0,30}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,1},{1,-5},30] (* Harvey P. Dale, Aug 13 2015 *)

[(-1)^n*fibonacci(3*n+2) for n in range(41)] # G. C. Greubel, Apr 20 2023

G.f.: (1-x)/(1+4*x-x^2).
a(n) = (sqrt(5)-2)^n * (1/2 - 3*sqrt(5)/10) + (-sqrt(5)-2)^n * (1/2 + 3*sqrt(5)/10).
a(n) = (-1)^n*Fibonacci(3*n+2).
a(n) = -4*a(n-1) + a(n-2), a(0)=1, a(1)=-5. - Philippe Deléham, Nov 03 2008
a(n) = (-1)^n*(A001076(n) + A001076(n+1)). - R. J. Mathar, Aug 10 2012
a(n) = (-1)^n*A015448(n+1). - R. J. Mathar, May 07 2019

cp's OEIS Frontend

A147703 Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

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A087206 a(n) = 2*a(n-1) + 4*a(n-2); with a(0)=1, a(1)=4.

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A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

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A152185 a(n) = -3*a(n-1) + 5*a(n-2), n > 1; a(0)=1, a(1)=-5.

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A099843 A transform of the Fibonacci numbers.

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A087206 a(n) = 2a(n-1) + 4a(n-2); with a(0)=1, a(1)=4.

A152185 a(n) = -3a(n-1) + 5a(n-2), n > 1; a(0)=1, a(1)=-5.