A054458 Convolution triangle based on A001333(n), n >= 1.
1, 3, 1, 7, 6, 1, 17, 23, 9, 1, 41, 76, 48, 12, 1, 99, 233, 204, 82, 15, 1, 239, 682, 765, 428, 125, 18, 1, 577, 1935, 2649, 1907, 775, 177, 21, 1, 1393, 5368, 8680, 7656, 4010, 1272, 238, 24, 1, 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1
Offset: 0
Examples
Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3. Triangle begins : 1 3, 1 7, 6, 1 17, 23, 9, 1 41, 76, 48, 12, 1 99, 233, 204, 82, 15, 1 239, 682, 765, 428, 125, 18, 1. - _Philippe Deléham_, Mar 25 2012 (0, 3, -2/3, -1/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins : 1 0, 1 0, 3, 1 0, 7, 6, 1 0, 17, 23, 9, 1 0, 41, 76, 48, 12, 1 0, 99, 233, 204, 82, 15, 1 0, 239, 682, 765, 428, 125, 15, 1. - _Philippe Deléham_, Mar 25 2012
Links
- Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Crossrefs
Formula
a(n, m) := ((n-m+1)*a(n, m-1) + (2n-m)*a(n-1, m-1) + (n-1)*a(n-2, m-1))/(4*m), n >= m >= 1; a(n, 0)= A001333(n+1); a(n, m) := 0 if n
G.f. for column m: LPell(x)*(x*LPell(x))^m, m >= 0, with LPell(x)= (1+x)/(1-2*x-x^2) = g.f. for A001333(n+1).
G.f.: (1+x)/(1-2*x-y*x-x^2-y*x^2). - Philippe Deléham, Mar 25 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012
A088210 Numerators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,...(n 2's)...,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], ...
1, 5, 17, 53, 157, 449, 1253, 3433, 9273, 24765, 65529, 172061, 448853, 1164409, 3006157, 7728337, 19794545, 50532469, 128621281, 326513669, 826887693, 2089505841, 5269572021, 13265211961, 33336792745, 83648953133, 209591807177
Offset: 0
Examples
a(3)/A088211(3) = [2;2,2,4] = 53/22.
References
- R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (See the foot of page 136.)
Links
- Paolo Xausa, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).
Programs
-
Mathematica
LinearRecurrence[{4, -2, -4, -1}, {1, 5, 17, 53}, 30] (* Paolo Xausa, Feb 08 2024 *)
Formula
G.f.: (1+x)(1-x^2)/(1-2*x-x^2)^2.
a(n) = A000129(n) + (n+1)*A000129(n+1) where A000129 are the Pell numbers. [Corrected by Paolo Xausa, Feb 08 2024]
A054460 A001333(n), n >= 1, convolved twice with itself.
1, 9, 48, 204, 765, 2649, 8680, 27312, 83313, 247985, 723624, 2077164, 5880797, 16454865, 45577200, 125130432, 340882113, 922265721, 2479938368, 6631802220, 17646603933, 46744464745, 123314065944, 324085913136, 848801213425
Offset: 0
Comments
a(n) = A054458(n+2,2) (third column of convolution triangle).
Links
- Index entries for linear recurrences with constant coefficients, signature (6,-9,-4,9,6,1)
Programs
-
Mathematica
LinearRecurrence[{6, -9, -4, 9, 6, 1}, {1, 9, 48, 204, 765, 2649}, 30] (* Paolo Xausa, Feb 06 2024 *)
Formula
a(n) = (n+1)*(LP2(n+1)+2*LP2(n)+LP2(n-1))/8, n >= 1, LP2(n) = A054459(n), a(0) = 1.
a(n) = (n+1)*((10*n+11)*LP(n)+(4*n+5)*LP(n-1))/16, n >= 1, LP(n) = A001333(n+1).
G.f.: ((1+x)/(1-2*x-x^2))^3.
a(n) = +6*a(n-1) -9*a(n-2) -4*a(n-3) +9*a(n-4) +6*a(n-5) +a(n-6). - R. J. Mathar, Feb 05 2024
A062134 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).
1, 2, 0, 8, 24, 16, 336, 832, 576, 128, 12480, 28480, 23680, 8960, 1280, 481920, 1208832, 1167360, 552960, 130560, 12288, 22786560, 61834752, 65709056, 35911680, 10895360, 1763328, 118784, 1280885760, 3645444096
Offset: 0
Comments
Examples
Triangle begins:
{1};
{2,0};
{8,24,16};
{336,832,576,128};
...
pPL1(1,n) = 1+2*n.
pPL2(1,n) = 2.
PL(1; n) = A054459(n) = ((1+2*n)*PL(n+1)+PL(n))/4.
Comments