A128099 -id:A128099 - cp's OEIS frontend

A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

1, 1, 10, 19, 109, 280, 1261, 3781, 15130, 49159, 185329, 627760, 2295721, 7945561, 28607050, 100117099, 357580549, 1258634440, 4476859381, 15804569341, 56096303770, 198337427839, 703204161769, 2488241012320, 8817078468241, 31211247579121, 110564953793290

Offset: 0

Olivier Gérard

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 10*a(n-2) equals the number of 10-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Borowska, L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=1, b=3.
Index entries for linear recurrences with constant coefficients, signature (1,9).

Cf. A015443, A015442, A026595, A128099.

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+9*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

m:=25; S:=series(1/(1-x-9*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 18 2020

CoefficientList[Series[1/(1-x-9*x^2), {x,0,25}], x] (* or *) LinearRecurrence[{1,9}, {1,1}, 25] (* G. C. Greubel, Apr 30 2017 *)

a(n)=([0,1; 9,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,1,-9) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = (((1+sqrt(37))/2)^(n+1) - ((1-sqrt(37))/2)^(n+1))/sqrt(37).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*9^k. - Paul Barry, Jul 20 2004
a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*3^(n-k)/2. - Paul Barry, Aug 28 2005
a(n) = Sum_{k=0..n} A109466(n,k)*(-9)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (-703*(1/2-sqrt(37)/2)^n + 199*sqrt(37)*(1/2-sqrt(37)/2)^n-333*(1/2+sqrt(37)/2)^n + 171*sqrt(37)*(1/2+sqrt(37)/2)^n)/(74*(5*sqrt(37)-14)). - Alexander R. Povolotsky, Oct 13 2010
a(n) = Sum_{k=1..n+1, k odd} C(n+1,k)*37^((k-1)/2)/2^n. - Vladimir Shevelev, Feb 05 2014
G.f.: 1/(1-x-9*x^2). - Philippe Deléham, Feb 19 2020
a(n) = J(n, 9/2), where J(n,x) are the Jacobsthal polynomials. - G. C. Greubel, Feb 18 2020
E.g.f.: exp(x/2)*(sqrt(37)*cosh(sqrt(37)*x/2) + sinh(sqrt(37)*x/2))/sqrt(37). - Stefano Spezia, Feb 19 2020

Edited by N. J. A. Sloane, Oct 11 2010

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

Original entry on oeis.org

1, 2, 4, 1, 8, 4, 16, 12, 1, 32, 32, 6, 64, 80, 24, 1, 128, 192, 80, 8, 256, 448, 240, 40, 1, 512, 1024, 672, 160, 10, 1024, 2304, 1792, 560, 60, 1, 2048, 5120, 4608, 1792, 280, 12, 4096, 11264, 11520, 5376, 1120, 84, 1, 8192, 24576, 28160, 15360

Offset: 1

Clark Kimberling, Feb 18 2012

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is the triangle given by (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and  along (first layer) skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.414213562373095... (A014176: Decimal expansion of the silver mean, 1+sqrt(2)), when n approaches infinity. - Zagros Lalo, Jul 31 2018

			First seven rows:
1
2
4...1
8...4
16..12..1
32..32..6
64..80..24..1
(2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
    1
    2,   0
    4,   1,  0
    8,   4,  0, 0
   16,  12,  1, 0, 0
   32,  32,  6, 0, 0, 0
   64,  80, 24, 1, 0, 0, 0
  128, 192, 80, 8, 0, 0, 0, 0

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
S. Halici, On some Pell polynomials , Acta Universitatis Apulensis, No. 29/2012, pp. 105-112.
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n

Cf. A028297, A207537, A133156, A038207, A099089.
Cf. A053117, A097610, A091894.
Cf. A013609, A038207.
Cf. A128099.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n -  k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ]  // Flatten (* Zagros Lalo, Jul 31 2018 *)

u(n,x) = u(n-1,x)+(x+1)*v(n-1,x), v(n,x) = u(n-1,x)+v(n-1,x), where u(1,x) = 1, v(1,x) = 1. Also, A207538 = |A133156|.
From Philippe Deléham, Mar 04 2012: (Start)
With 0<=k<=n:
Mirror image of triangle in A099089.
Skew version of A038207.
Riordan array (1/(1-2*x), x^2/(1-2*x)).
G.f.: 1/(1-2*x-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190958(n+1), A127357(n), A090591(n), A089181(n+1), A088139(n+1), A045873(n+1), A088138(n+1), A088137(n+1), A099087(n), A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively.
T(n,k) = 2*T(n-1,k) + T(-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n, k) = 0 if k<0 or if k>n. (End)
T(n,k) = A013609(n-k, n-2*k+1). - Johannes W. Meijer, Sep 05 2013
From Tom Copeland, Feb 11 2016: (Start)
A053117 is a reflected, aerated and signed version of this entry. This entry belongs to a family discussed in A097610 with parameters h1 = -2 and h2 = -y.
Shifted o.g.f.: G(x,t) = x / (1 - 2 x - t x^2).
The compositional inverse of G(x,t) is Ginv(x,t) = -[(1 + 2x) - sqrt[(1+2x)^2 + 4t x^2]] / (2tx) = x - 2 x^2 + (4-t) x^3 - (8-6t) x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0).
(End)

A052942 Expansion of 1/((1+x)(1-2x+2x^2-2x^3)).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 7, 9, 15, 25, 39, 57, 87, 137, 215, 329, 503, 777, 1207, 1865, 2871, 4425, 6839, 10569, 16311, 25161, 38839, 59977, 92599, 142921, 220599, 340553, 525751, 811593, 1252791, 1933897, 2985399, 4608585, 7114167, 10981961

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 4, 3*a(n-4) equals the number of 3-colored compositions of n with all parts >= 4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

a(n+3) equals the number of ternary words of length n having at least 3 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 933
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2).

Column k=3 of A143453.

a:=[1,1,1,1];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-4]; od; a; # G. C. Greubel, Jun 12 2019

I:=[1,1,1,1]; [n le 4 select I[n] else Self(n-1)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 10 2015

spec := [S,{S=Sequence(Union(Z,Prod(Union(Z,Z),Z,Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(add(binomial(n-3*k,k)*2^k, k=0..floor(n/3)), n=0..39); # Zerinvary Lajos, Apr 03 2007
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Sequence(Z, card >3)}, unlabeled]: seq(count(SeqSeqSeqL, size=j+4), j=0..39); # Zerinvary Lajos, Apr 04 2009
a := n -> `if`(n<9, [1, 1, 1, 1, 3, 5, 7, 9, 15][n+1], hypergeom([(1-n)/4,(2-n)/4,(3-n)/4,-n/4], [(1-n)/3,(2-n)/3,-n/3], -512/27)):
seq(simplify(a(n)),n=0..39); # Peter Luschny, Mar 09 2015

CoefficientList[Series[1/(1-x-2*x^4), {x,0,40}], x] (* Vincenzo Librandi, Mar 10 2015 *)
LinearRecurrence[{1,0,0,2},{1,1,1,1},50] (* Harvey P. Dale, Aug 17 2024 *)

Vec( 1/(1-x-2*x^4) + O(x^66)) \\ Joerg Arndt, Aug 28 2013

(1/(1-x-2*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

G.f.: 1/(1-x-2*x^4).
a(n) = a(n-1) + 2*a(n-4), with a(1)=1, a(0)=1, a(2)=1, a(3)=1.
a(n) = Sum_{alpha=RootOf(-1+_Z+2*_Z^4)} (1/539)*(27 + 72*alpha^3 + 96*alpha^2 + 128*alpha)*alpha^(-1-n)).
a(n) = Sum_{k=0..floor(n/3)} A128099(n-2*k, k). - Johannes W. Meijer, Aug 28 2013
a(n) = hypergeom([(1-n)/4,(2-n)/4,(3-n)/4,-n/4],[(1-n)/3,(2-n)/3,-n/3],-512/27) for n>=9. - Peter Luschny, Mar 09 2015

More terms from James Sellers, Jun 06 2000

A077909 Expansion of 1/((1-x)(1+x+x^2+2x^3)).

Original entry on oeis.org

1, 0, 0, -1, 2, 0, 1, -4, 4, -1, 6, -12, 9, -8, 24, -33, 26, -40, 81, -92, 92, -161, 254, -276, 345, -576, 784, -897, 1266, -1936, 2465, -3060, 4468, -6337, 7990, -10588, 15273, -20664, 26568, -36449, 51210, -67896, 89585, -124108, 170316, -225377, 303278, -418532, 566009, -754032, 1025088

Offset: 0

N. J. A. Sloane, Nov 17 2002

The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
  .___.     ._____.  ._____.
  | .. |     | .. | |  | | ._. |
  || ||     || || |  | || ||
  |. .|  ,  | .| .|  |. |. |
  | || |     | | || |  | |_| | |
  |___|     ||____|  |___|_|. - Alois P. Heinz, Jan 03 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,0,-1,2)

Partial sums of A077976.

Cf. A174249, A233427, A234312, A234931, A247126.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
        <<1, 0, 0, -1>>)[1, 1]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 20 2013

CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* Jean-François Alcover, Jun 08 2015, after Arkadiusz Wesolowski *)

Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ Joerg Arndt, Aug 28 2013

a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - Johannes W. Meijer, Aug 28 2013

G.f.: 1/(1 + x^3 - 2*x^4). - Arkadiusz Wesolowski, Nov 20 2013

A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1

Offset: 0

Gary W. Adamson, Feb 14 2010

The row sums equal A052952.
Let the triangle = M. Then lim_{n->infinity} M^n = A173285 as a left-shifted vector.
A173284 * [1, 2, 3, ...] = A054451: (1, 1, 4, 5, 12, 17, 33, ...). - Gary W. Adamson, Mar 03 2010
From Johannes W. Meijer, Sep 05 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A104762.
The diagonal sums lead to A004695. (End)

			First few rows of the triangle:
    1;
    1;
    2,   1;
    3,   1;
    5,   2,  1;
    8,   3,  1;
   13,   5,  2,  1;
   21,   8,  3,  1;
   34,  13,  5,  2,  1;
   55,  21,  8,  3,  1;
   89,  34, 13,  5,  2, 1;
  144,  55, 21,  8,  3, 1;
  233,  89, 34, 13,  5, 2, 1;
  377, 144, 55, 21,  8, 3, 1;
  610, 233, 89, 34, 13, 5, 2, 1;
  ...

Cf. A000045, A004695, A052952, A054451, A173285.

Cf. (Similar triangles) A008315 (Catalan), A011973 (Pascal), A102541 (Losanitsch), A122196 (Fractal), A122197 (Fractal), A128099 (Pell-Jacobsthal), A152198, A152204, A207538, A209634.

T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013

Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
From Johannes W. Meijer, Sep 05 2013: (Start)
T(n,k) = A000045(n-2*k+1), n >= 0 and 0 <= k <= floor(n/2).
T(n,k) = A104762(n-k, k). (End)

Term a(15) corrected by Johannes W. Meijer, Sep 05 2013

A305098 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -1, 1, 2, -1, -4, 1, 6, 4, -1, -8, -12, 1, 10, 24, 8, -1, -12, -40, -32, 1, 14, 60, 80, 16, -1, -16, -84, -160, -80, 1, 18, 112, 280, 240, 32, -1, -20, -144, -448, -560, -192, 1, 22, 180, 672, 1120, 672, 64, -1, -24, -220, -960, -2016, -1792, -448

Offset: 0

Shara Lalo, May 25 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2*x)^n).
The coefficients in the expansion of 1/(1+x-2x^2) are given by the sequence generated by the row sums.
When n is even the numbers in the row are positive, and when n is odd the numbers in the row are negative.

			Triangle begins:
   1;
  -1;
   1,   2;
  -1,  -4;
   1,   6,    4;
  -1,  -8,  -12;
   1,  10,   24,     8;
  -1, -12,  -40,   -32;
   1,  14,   60,    80,     16;
  -1, -16,  -84,  -160,    -80;
   1,  18,  112,   280,    240,     32;
  -1, -20, -144,  -448,   -560,   -192;
   1,  22,  180,   672,   1120,    672,     64;
  -1, -24, -220,  -960,  -2016,  -1792,   -448;
   1,  26,  264,  1320,   3360,   4032,   1792,    128;
  -1, -28, -312, -1760,  -5280,  -8064,  -5376,  -1024;
   1,  30,  364,  2288,   7920,  14784,  13440,   4608,   256;
  -1, -32, -420, -2912, -11440, -25344, -29568, -15360, -2304;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.

Signed version of A128099.
Row sums give A077925.
Cf. A303872, A033999 (column 0).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, -t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

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

A113726 A Jacobsthal convolution.

Original entry on oeis.org

1, 0, 1, 4, 5, 8, 25, 44, 77, 176, 353, 660, 1365, 2776, 5417, 10876, 21981, 43648, 87153, 175076, 349669, 698280, 1398585, 2797260, 5590381, 11184720, 22373761, 44735284, 89474165, 178969208, 357910345, 715807004, 1431683837, 2863325216

Offset: 0

Paul Barry, Nov 08 2005

Convolution of A001045(n+1) and A001607(n+1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,4,4).

Cf. A094686, A089977, A001045, A001607.

LinearRecurrence[{0,1,4,4},{1,0,1,4},40] (* Harvey P. Dale, Apr 30 2025 *)

G.f.: 1/((1-x-2*x^2)*(1+x+2*x^2)).
a(n) = a(n-2) + 4*a(n-3) + 4*a(n-4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*2^k*(1+(-1)^(n-k))/2.
a(n) = 2^n/3 + (-1)^n/6 + A001607(n+1)/2. - R. J. Mathar, Aug 23 2011
a(n) = sum(A128099(n, n-2*k), k=0..floor(n/2)). - Johannes W. Meijer, Aug 28 2013

A014291 Imaginary Rabbits: imaginary part of a(0)=i; a(1)=-i; a(n) = a(n-1) + i*a(n-2), with i = sqrt(-1).

Original entry on oeis.org

1, -1, -1, -1, -2, -2, -1, 1, 5, 11, 18, 24, 25, 15, -13, -65, -142, -234, -313, -327, -199, 163, 838, 1840, 3041, 4079, 4279, 2639, -2042, -10802, -23841, -39519, -53155, -55989, -34982, 25544, 139225, 308895, 513547, 692655

Offset: 0

Olivier Gérard

Second differences give -a(n-2).

Charles Gely, "Lapins imaginaires et valeurs propres". Quadrature Quarterly #30.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-1).

Cf. A128099.

a:=[1,-1,-1,-1];; for n in [5..50] do a[n]:=2*a[n-1]-a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jun 13 2019

I:=[1, -1, -1, -1]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) - Self(n-4): n in [1..50]]; // Vincenzo Librandi, Oct 23 2012

CoefficientList[Series[(2*x-1)*(x-1)/(x^4+x^2-2*x+1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 23 2012 *)

my(x='x+O('x^50)); Vec((1-x)*(1-2*x)/(1-2*x+x^2+x^4)) \\ G. C. Greubel, Jun 13 2019

((1-x)*(1-2*x)/(1-2*x+x^2+x^4)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019

a(n) = 2*a(n-1) - a(n-2) - a(n-4).
G.f.: (1-x)*(1-2*x)/(1-2*x+x^2+x^4).
a(n) = Im( -i*J(n, -i) + J(n-1, -i) ), where J(n,x) is the Jacobsthal polynomial, whose coefficient array is A128099, and i = sqrt(-1). - G. C. Greubel, Jun 15 2019

A095977 Expansion of g.f. 2x / ((1+x)^2(1-2*x)^2).

Original entry on oeis.org

2, 4, 14, 32, 82, 188, 438, 984, 2202, 4852, 10622, 23056, 49762, 106796, 228166, 485448, 1029162, 2174820, 4582670, 9631360, 20194802, 42253724, 88235734, 183927992, 382769082, 795364308, 1650380958, 3420066544, 7078742402, 14634703372, 30223843942, 62356562216

Offset: 1

Ralf Stephan, Jul 16 2004

Number of 2 X 2 tiles in all tilings of a 3 X (n+1) rectangle with 1 X 1 and 2 X 2 square tiles. - Emeric Deutsch, Feb 18 2007

The terms of this sequence have a primitive divisor for all terms beyond the 4th if and only if n is not of the form 4k+2, for some nonnegative integer k. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5
A. Flatters, Prime divisors of some Lehmer-Pierce sequences, arXiv:0708.2190 [math.NT], 2007.
R. P. Grimaldi, Tilings, Compositions, and Generalizations, J. Int. Seq. 13 (2010), 10.6.5, page 7.
Luka Podrug, Horadam cubes, arXiv:2410.03193 [math.CO], 2024. See p. 11.
Helmut Prodinger, On binary representations of integers with digits -1,0,1, Integers 0 (2000), #A08.
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A128099, A073371.

a:=n->n/9*2^(n+2)+1/27*2^(n+3)-2*n/9*(-1)^n-8/27*(-1)^n: seq(a(n),n=1..30); # Emeric Deutsch, Feb 18 2007

Table[(1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n) , {n,1,50}] (* G. C. Greubel, Dec 28 2016 *)

Vec(2*x / ((1+x)^2 * (1-2*x)^2) + O(x^50)) \\ Michel Marcus, Nov 07 2015

a(n) = (1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n).
a(n) = 2 * A073371(n-1).
a(n) = Sum_{k=0..floor((n+1)/2)} k*2^k*binomial(n+1-k,k). - Emeric Deutsch, Feb 18 2007
E.g.f.: 2*(cosh(x/2) + sinh(x/2))*(15*x*cosh(3*x/2) + (8 + 9*x)*sinh(3*x/2))/27. - Stefano Spezia, Oct 12 2024

A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

Original entry on oeis.org

1, 4, 7, 1, 10, 4, 13, 7, 1, 16, 10, 4, 19, 13, 7, 1, 22, 16, 10, 4, 25, 19, 13, 7, 1, 28, 22, 16, 10, 4, 31, 25, 19, 13, 7, 1, 34, 28, 22, 16, 10, 4, 37, 31, 25, 19, 13, 7, 1, 40, 34, 28, 22, 16, 10, 4, 43, 37, 31, 25, 19, 13, 7, 1, 46, 40, 34, 28, 22, 16, 10

Offset: 1

Ctibor O. Zizka, Mar 11 2012

OEIS contains a lot of similar sequences, for example A152204, A122196, A173284.
Row sums for this sequence gives A006578.
In general, by given triangle with (A-B,2*A-B,...,A*n-B,...) in every column, shifted down K-times, we have the row sum s(n)= A*(n*n+K*n+nmodK)/(2*K) - B*(n+nmodK)/K. In this sequence K=2,A=3,B=2, in A152204 K=2,A=2,B=1.
No triangle with primes in every column, shifted down by K>=2 in OEIS, no row sums of it in OEIS.
From Johannes W. Meijer, Sep 28 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A143971.
The alternating row sums equal A004524(n+2) + 2*A004524(n+1).
The antidiagonal sums equal A171452(n+1). (End)

			Triangle:
1
4
7,  1
10, 4
13, 7,  1
16, 10, 4
19, 13, 7,  1
22, 16, 10, 4
25, 19, 13, 7,  1
28, 22, 16, 10, 4
...

Cf. A008315, A011973, A102541, A122196, A122197, A128099, A152198, A152204, A173284, A207538.

Cf. (Related to triangle sums) A006578, A000217, A002620, A004524, A171452.

T := (n, k) -> 3*n - 6*k + 4: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..15); # Johannes W. Meijer, Sep 28 2013

From Johannes W. Meijer, Sep 28 2013: (Start)
T(n, k) = 3*n - 6*k + 4, n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A143971(n-k+1, k), n >= 1 and 1 <= k <= floor((n+1)/2). (End)

cp's OEIS Frontend

A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

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A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

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A052942 Expansion of 1/((1+x)*(1-2*x+2*x^2-2*x^3)).

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A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).

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A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

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A305098 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A113726 A Jacobsthal convolution.

A052942 Expansion of 1/((1+x)(1-2x+2x^2-2x^3)).

A077909 Expansion of 1/((1-x)(1+x+x^2+2x^3)).

A095977 Expansion of g.f. 2x / ((1+x)^2(1-2*x)^2).