A128915 -id:A128915 - cp's OEIS frontend

A127836 Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^(n-1)*P_{n-2}(x).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5

Offset: 0

N. J. A. Sloane, Apr 07 2007

P_n(x) has degree A002620(n).
Row sums are the Fibonacci numbers (A000045). - Emeric Deutsch, May 12 2007
T(n,k) is the number of Fibonacci words of length n-1 in which the sum of the positions of the 0's is equal to k. A Fibonacci binary word is a binary word having no 00 subword. Examples: T(5,4) = 2 because we have 1110 and 0101; T(7,6) = 3 because we have 111110, 101011 and 011101. - Emeric Deutsch, Jan 04 2009

			Triangle begins:
   1;
   1;
   1, 1;
   1, 1, 1;
   1, 1, 1, 1, 1;
   1, 1, 1, 1, 2, 1, 1;
   1, 1, 1, 1, 2, 2, 2, 1, 1, 1;
   1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1;
   1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1;
   1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1;
   ...

Seiichi Manyama, Table of n, a(n) for n = 0..9548 (rows n=0..48 of triangle, flattened).
M. Barnabei, F. Bonetti, S. Elizalde, M. Silimbani, Descent sets on 321-avoiding involutions and hook decompositions of partitions, arXiv preprint arXiv:1401.3011 [math.CO], 2014.
A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp. See Identity 3-18, pp. 26-27.
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Rows converge to A003114 (coefficients in expansion of the first Rogers-Ramanujan identities). Cf. A128915, A119469.

P[0]:=1; P[1]:=1; d:=[0,0]; M:=14; for n from 2 to M do P[n]:=expand(P[n-1]+q^(n-1)*P[n-2]);
lprint(seriestolist(series(P[n],q,M^2))); d:=[op(d),degree(P[n],q)]; od: d;

P[0] = P[1] = 1; P[n_] := P[n] = P[n-1] + x^(n-1) P[n-2];
Table[CoefficientList[P[n], x], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 23 2018 *)

P(n, x) := if n = 0 or n = 1 then 1 else P(n - 1, x) + x^(n  - 1)*P(n - 2, x)$ create_list(ratcoef(expand(P(n, x)), x, k), n, 0, 10, k, 0, floor(n^2/4)); /* Franck Maminirina Ramaharo, Nov 30 2018 */

A279543 a(n) = a(n-1) + 3^n * a(n-2) with a(0) = 1 and a(1) = 1.

Original entry on oeis.org

1, 1, 10, 37, 847, 9838, 627301, 22143007, 4137864868, 439978671649, 244776761262181, 78185678507867584, 130162592460442600405, 124783388108159412726037, 622688428086038843429228482, 1791127919536971393223950620041

Offset: 0

Seiichi Manyama, Dec 31 2016

nonn

The Rogers-Ramanujan continued fraction is defined by R(q) = q^(1/5)/(1+q/(1+q^2/(1+q^3/(1+ ... )))). The limit of a(n)/A015460(n+2) is 3^(-1/5) * R(3).

			1/1 = a(0)/A015460(2).
1/(1+3/1) = 1/4 = a(1)/A015460(3).
1/(1+3/(1+3^2/1)) = 10/13 = a(2)/A015460(4).
1/(1+3/(1+3^2/(1+3^3/1))) = 37/121 = a(3)/A015460(5).

Seiichi Manyama, Table of n, a(n) for n = 0..90
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A015460, A128915.

Cf. similar sequences with the recurrence a(n-1) + q^n * a(n-2) for n>1, a(0)=1 and a(1)=1: A280294 (q=2), this sequence (q=3), A280340 (q=10).

RecurrenceTable[{a[n] == a[n - 1] + 3^n*a[n - 2], a[0] == 1, a[1] == 1}, a, {n, 15}] (* Michael De Vlieger, Dec 31 2016 *)

A280294 a(n) = a(n-1) + 2^n * a(n-2) with a(0) = 1 and a(1) = 1.

Original entry on oeis.org

1, 1, 5, 13, 93, 509, 6461, 71613, 1725629, 38391485, 1805435581, 80431196861, 7475495336637, 666367860021949, 123144883455482557, 21958686920654707389, 8092381769059159562941, 2886261393833112966453949, 2124255587862077437434059453

Offset: 0

Seiichi Manyama, Dec 31 2016

nonn

The Rogers-Ramanujan continued fraction is defined by R(q) = q^(1/5)/(1+q/(1+q^2/(1+q^3/(1+ ... )))). The limit of a(n)/A015459(n+2) is 2^(-1/5) * R(2).

			1/1 = a(0)/A015459(2).
1/(1+2/1) = 1/3 = a(1)/A015459(3).
1/(1+2/(1+2^2/1)) = 5/7 = a(2)/A015459(4).
1/(1+2/(1+2^2/(1+2^3/1))) = 13/31 = a(3)/A015459(5).

Seiichi Manyama, Table of n, a(n) for n = 0..114
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A015463, A015459, A128915.

Cf. similar sequences with the recurrence a(n-1) + q^n * a(n-2) for n>1, a(0)=1 and a(1)=1: this sequence (q=2), A279543 (q=3), A280340 (q=10).

nxt[{n_,a_,b_}]:={n+1,b,b+2^(n+1)*a}; NestList[nxt,{1,1,1},20][[All,2]] (* Harvey P. Dale, Jul 17 2020 *)

def a():
    a, b, p = 1, 0, 1
    while True:
        p, a, b = p + p, b, b + p * a
        yield b
A280294 = a()
print([next(A280294) for  in range(19)]) # _Peter Luschny, Dec 05 2017

A280340 a(n) = a(n-1) + 10^n * a(n-2) with a(0) = 1 and a(1) = 1.

Original entry on oeis.org

1, 1, 101, 1101, 1011101, 111111101, 1011212111101, 1112122222111101, 101122323232322111101, 1112223344434333322111101, 1011224344546565545343322111101, 111223345667777878776655443322111101, 1011224455769911213121200887756443322111101

Offset: 0

Seiichi Manyama, Dec 31 2016

nonn

The Rogers-Ramanujan continued fraction is defined by R(q) = q^(1/5)/(1+q/(1+q^2/(1+q^3/(1+ ... )))). The limit of a(n)/A015468(n+2) is 10^(-1/5) * R(10).

a(n) has A004652(n+1) digits. The last n digits are the same as the last n digits of a(n-1). - Robert Israel, Jan 12 2017

			1/1 = a(0)/A015468(2).
1/(1+10/1) = 1/11 = a(1)/A015468(3).
1/(1+10/(1+10^2/1)) = 101/111 = a(2)/A015468(4).
1/(1+10/(1+10^2/(1+10^3/1))) = 1101/11111 = a(3)/A015468(5).

Seiichi Manyama, Table of n, a(n) for n = 0..62
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A004652, A015468, A128915.

Cf. similar sequences with the recurrence a(n-1) + q^n * a(n-2) for n>1, a(0)=1 and a(1)=1: A280294 (q=2), A279543 (q=3), this sequence (q=10).

A[0]:= 1: A[1]:= 1:
for n from 2 to 20 do A[n]:= A[n-1]+10^n*A[n-2] od:
seq(A[i],i=0..20); # Robert Israel, Jan 12 2017

RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+10^n a[n-2]},a,{n,15}] (* Harvey P. Dale, Jul 12 2020 *)

a(n) a(n-3) = 10 a(n-2) a(n-1) - 10 a(n-2)^2 + a(n-1) a(n-3). - Robert Israel, Jan 12 2017

cp's OEIS Frontend

A127836 Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^(n-1)*P_{n-2}(x).

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A279543 a(n) = a(n-1) + 3^n * a(n-2) with a(0) = 1 and a(1) = 1.

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A280294 a(n) = a(n-1) + 2^n * a(n-2) with a(0) = 1 and a(1) = 1.

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A280340 a(n) = a(n-1) + 10^n * a(n-2) with a(0) = 1 and a(1) = 1.

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