editing

approved

Number of Carlitz compositions of n (see A003242) such that the first and last parts are equal.

proposed

editing

editing

proposed

C_x(N) = { my(g =1/(1-sum(k=1, N, x^k/(1+x^k)))); g}

Cf. A001045, A003242, A285981, A292200, A291941, A374517.

Number of Carlitz compositions of n such that the first part and last part parts are equal.

A_x(i, N) = {my( x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2); f}

D_x(2040)

Cf. A001045, A003242, A285981, A292200, A374517.

allocated for John Tyler RascoeNumber of Carlitz compositions of n such that the first part and last part are equal.

1, 1, 1, 1, 2, 3, 2, 7, 11, 17, 26, 54, 86, 155, 272, 464, 816, 1447, 2507, 4400, 7706, 13456, 23570, 41293, 72212, 126394, 221282, 387219, 677714, 1186311, 2076170, 3633761, 6360219, 11131698, 19483066, 34100455, 59683664, 104460655, 182832044, 319999739

0,5

G.f.: 1 + Sum_{i>0} (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2 where C(x) is the g.f. for A003242.

a(7) = 7 counts: (1,2,1,2,1), (1,2,3,1), (1,3,2,1), (1,5,1), (2,3,2), (3,1,3), and (7).

(PARI)

C_x(N) = { my(g =1/(1-sum(k=1, N, x^k/(1+x^k)))); g}

A_x(i, N) = {my( f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2); f}

D_x(N) = {my( x='x+O('x^N), f=1+sum(i=1, N, A_x(i, N))); Vec(f)}

D_x(20)

Cf. A001045, A003242, A374517.

allocated

nonn,easy

John Tyler Rascoe, Aug 16 2024

approved

editing

allocated for John Tyler Rascoe

recycled

allocated

editing

approved

a(2*n) = A135318(n), a(2*n+1) = A112387(n).

1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 3, 5, 8, 8, 5, 11, 16, 16, 11, 21, 32, 32, 21, 43, 64, 64, 43, 85, 128, 128, 85, 171, 256, 256, 171, 341, 512, 512, 341, 683, 1024, 1024, 683, 1365, 2048, 2048, 1365, 2731, 4096, 4096, 2731, 5461, 8192, 8192, 5461, 10923, 16384

0,6

<a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,0,0,0,2).

a(4*n+1) + a(4*n+1) + a(4*n+2) + a(4*n+3) = 4, 6, 14, 26, ... = A084214(n+2) = 2*A048573(n).

a(2*n) + a(2*n+1) = 2, 2, 3, 3, 7, 7, ... = repeat A048573(n).

a(3*n) + a(3*n+1) + a(3*n+2) = 3, 4, 6, 11, 21, 32, 48, 85, 171, 256, 384 ... = b(n). 3, 11, 21, 85, 171, ... are from A001045. b(2*n) is divisible by 3.

Cf. A048573, A084214, A112387, A135318.

Cf. A001045.

nonn,easy,changed

recycled

Paul Curtz, Aug 03 2024

proposed

editing