A000621 -id:A000621 - cp's OEIS frontend

A349365 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * A(x^2)).

1, 2, 4, 10, 24, 60, 148, 370, 920, 2296, 5720, 14268, 35568, 88700, 221156, 551482, 1375096, 3428888, 8549944, 21319624, 53160896, 132558360, 330537528, 824204780, 2055176304, 5124638944, 12778424976, 31863351980, 79452130896, 198116051644, 494007751668

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000621, A127680.

Cf. A218032, A319436.

nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x - x A[x^2]) + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 2] + Sum[a[Floor[k/2]] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 30}]

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, (i-1)\2, v[j+1]*v[i-2*j])); v; \\ Seiichi Manyama, Nov 26 2023

G.f.: 1 / (1 - x - x / (1 - x^2 - x^2 / (1 - x^4 - x^4 / (1 - x^8 - x^8 / (1 - ...))))).
a(0) = 1, a(1) = 2; a(n) = a(n-2) + Sum_{k=0..n-1} a(floor(k/2)) * a(n-k-1).
a(n) ~ c * d^n, where d = 2.4935271724548067876965033643037290636931200352851874903211458249308... and c = 0.6156170089558875346518987360369130661426312977478830077668203229773... - Vaclav Kotesovec, Nov 16 2021
a(0) = 1; a(n) = a(n-1) + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-1-2*k). - Seiichi Manyama, Nov 26 2023

A206743 G.f.: 1/(1 - x/(1 - x^2/(1 - x^5/(1 - x^12/(1 - x^29/(1 - x^70/(1 -...- x^Pell(n)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 99, 164, 272, 450, 746, 1235, 2046, 3389, 5613, 9299, 15402, 25514, 42262, 70005, 115962, 192084, 318182, 527053, 873043, 1446161, 2395504, 3968060, 6572925, 10887788, 18035177, 29874537, 49485965, 81971484, 135782448

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

From Clark Kimberling, Jun 12 2016: (Start)
Number of real integers in n-th generation of tree T(2i) defined as follows.
Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
For r = 2i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
See A274142 for a guide to related sequences. (End)

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 +...

Kenny Lau, Table of n, a(n) for n = 0..1000

Cf. A000621, A206741, A274142.

A206743 := proc(r)
local gs,n,gs2,el,a ;
gs := [2,r] ;
for n from 3 do
gs2 := [] ;
for el in gs do
gs2 := [op(gs2),el+1,r*el] ;
end do:
gs := gs2 ;
a := 0 ;
for el in gs do
if type(el,'realcons') then
a := a+1 :
end if;
end do:
print(n,a) ;
end do:
end proc: # R. J. Mathar, Jun 16 2016

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]]; u = Table[t[[k]] /. x -> 2 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}] (* Clark Kimberling, Jun 12 2016 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=local(CF=1+x*O(x^n),M=ceil(log(2*n+1)/log(2.4))); for(k=0, M, CF=1/(1-x^Pell(M-k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

N = 1000
pell = [0,1]
c = 2
while c < N:
    pell.append(c)
    c = pell[-1]*2 + pell[-2]
pell.reverse()
gf = [0]*(N+1)
for p in pell:
    gf = [-x for x in gf]
    gf[0] += 1
    quotient = [0]*(N+1)
    remainder = [0]*(N+1)
    remainder[p] = 1
    for n in range(N+1):
        q = remainder[n]//gf[0]
        for i in range(n,N+1):
            remainder[i] -= q*gf[i-n]
        quotient[n] = q
    gf = quotient
for i in range(N+1):
    print(i,gf[i])
# Kenny Lau, Aug 01 2017

a(n) ~ c * d^n, where d = 1.6564594309887754808836889708489581749625897572527517021957723319642053908... and c = 0.3844078703275069072126260832303344589497793302955451672191630264983... - Vaclav Kotesovec, Aug 25 2017

A351972 a(n) = 1 + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 40, 78, 151, 294, 572, 1115, 2172, 4234, 8252, 16088, 31361, 61140, 119191, 232370, 453010, 883167, 1721768, 3356675, 6543988, 12757830, 24871992, 48489172, 94531974, 184294706, 359291464, 700456240, 1365573493, 2662252082, 5190190005

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Cf. A000621, A007317, A351973.

a[n_] := a[n] = 1 + Sum[a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 34}]
nmax = 34; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x^2])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x^2))).

A351973 a(n) = (-1)^n + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 3, 4, 6, 9, 12, 18, 24, 36, 49, 72, 99, 144, 200, 289, 404, 581, 816, 1168, 1646, 2350, 3320, 4730, 6692, 9522, 13487, 19174, 27177, 38614, 54757, 77771, 110318, 156646, 222246, 315526, 447719, 635569, 901924, 1280257, 1816886, 2578911

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Cf. A000621, A005043, A351972.

a[n_] := a[n] = (-1)^n + Sum[a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 47}]
nmax = 47; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x^2])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x^2))).

A253190 Triangle T(n, m)=Sum_{k=1..(n-m)/2} C(m+k-1, k)*T((n-m)/2, k), T(n,n)=1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 2, 0, 6, 0, 5, 0, 1, 0, 6, 0, 10, 0, 6, 0, 1, 3, 0, 13, 0, 15, 0, 7, 0, 1, 0, 11, 0, 24, 0, 21, 0, 8, 0, 1, 5, 0, 27, 0, 40, 0, 28, 0, 9, 0, 1, 0, 20, 0, 55, 0, 62, 0, 36, 0, 10, 0, 1

Offset: 1

Vladimir Kruchinin, Mar 24 2015

			1;
0, 1;
1, 0, 1;
0, 2, 0, 1;
1, 0, 3, 0, 1;
0, 3, 0, 4, 0, 1;
2, 0, 6, 0, 5, 0, 1;

Cf. A000621 (row sums), A003600, A253184, A253189.

A253190 := proc(n,m)
    option remember;
    if n = m then
        1;
    elif type(n-m,'odd') then
        0 ;
    else
        add(binomial(m+k-1,k)*procname((n-m)/2,k),k=1..(n-m)/2) ;
    end if;
end proc: # R. J. Mathar, Dec 16 2015

T(n, m):=if n=m then 1 else sum(binomial(m+k-1, k)*T((n-m)/2, k), k, 1, (n-m)/2);

G.f.: A(x)^m=Sum_{n>=m} T(n,m)x^n, A(x)=Sum_{n>0} a(n)*x^(2*n-1), a(n) - is A000621.

A367637 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^6)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 43, 55, 71, 92, 119, 153, 196, 251, 322, 414, 533, 686, 882, 1133, 1455, 1869, 2402, 3088, 3970, 5103, 6558, 8427, 10829, 13917, 17888, 22992, 29551, 37979, 48809, 62727, 80617, 103612, 133167

Offset: 0

Seiichi Manyama, Dec 01 2023

nonn

This sequence is different from A005708.

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000621, A367794, A367800.

Cf. A005708, A101916.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\6, v[j+1]*v[i-6*j])); v;

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(k) * a(n-1-6*k).

A367794 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 346, 478, 660, 911, 1259, 1740, 2404, 3320, 4586, 6336, 8754, 12093, 16705, 23077, 31881, 44043, 60844, 84053, 116116, 160410, 221602, 306136, 422916, 584242, 807110, 1114996

Offset: 0

Seiichi Manyama, Nov 30 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000108, A000621, A367637, A367800.

Cf. A367653, A367692, A367694, A367720.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\4, v[j+1]*v[i-4*j])); v;

from functools import lru_cache
@lru_cache(maxsize=None)
def A367794(n): return sum(A367794(k)*A367794(n-1-(k<<2)) for k in range(n+3>>2)) if n else 1 # Chai Wah Wu, Nov 30 2023

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} a(k) * a(n-1-4*k).

A367800 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^8)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623

Offset: 0

Seiichi Manyama, Dec 01 2023

nonn

a(n) = A005710(n-1) up to n=72, but then the two sequences start to differ. - R. J. Mathar, Dec 04 2023

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000621, A367794, A367637.

Cf. A005710, A101918.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\8, v[j+1]*v[i-8*j])); v;

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/8)} a(k) * a(n-1-8*k).

A036675 G.f. satisfies A(x) = 1 + xA(x)^2A(x^2).

Original entry on oeis.org

1, 1, 2, 6, 18, 59, 198, 690, 2450, 8878, 32632, 121518, 457262, 1736526, 6646340, 25613086, 99298674, 387021728, 1515594560, 5960406102, 23530528512, 93216984177, 370450977206, 1476458287082, 5900150928510, 23635544130948

Offset: 0

N. J. A. Sloane

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000621, A101913

A := 1; f := proc(n) global A; coeff(series( 1+x*(A*subs(x=x^2,A)), x, n+1), x,n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;

terms = 26; A[] = 0; Do[A[x] = 1 + x*A[x]^2*A[x^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)

T(n,m):=if m=n then 1 else sum((m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m);
makelist(T(2*n+1,1),n,0,30); /* Vladimir Kruchinin, Mar 18 2015 */

a(n)=local(A,m); if(n<0,0,m=2; A=1+O(x); while(m<=n+1,m*=2; A=2/(1+sqrt(1-4*x*subst(A,x,x^2)))); polcoeff(A,n))

G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x^2/(1-x^2)) (continued fraction); more generally g.f. C(x/(1-x^2/(1-x^2))) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011]
a(n) ~ c * d^n / n^(3/2), where d = 4.250770453055989899189676464071962617426..., c = 0.600960911911396921862654605015399962... . - Vaclav Kotesovec, Aug 10 2014
a(n) = T(2*n+1,1), where T(n,m) = sum(i=1..n-m, (m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i)), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015

A036676 Used by Polya in calculating A000598.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 8, 20, 46, 102, 220, 461, 948, 1921, 3836, 7574, 14810, 28705, 55212, 105485, 200300, 378262, 710795, 1329603, 2476817, 4596297, 8499396, 15665681, 28786696, 52747907, 96398485, 175735593, 319623891, 580054269

Offset: 0

N. J. A. Sloane

nonn

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; q^{I}(x) on p. 441.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; q^{I}(x) on p. 441. (Annotated scanned copy)

G.f.: x * q(x) * (q(x)^3 - 3*q(x)*q(x^2) + 2*q(x^3)) / 6 where q(x) is the g.f. of A000621 with offset 0 [from Polya]. - Sean A. Irvine, Nov 21 2020

a(33) corrected by Sean A. Irvine, Nov 21 2020

cp's OEIS Frontend

A349365 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * A(x^2)).

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A206743 G.f.: 1/(1 - x/(1 - x^2/(1 - x^5/(1 - x^12/(1 - x^29/(1 - x^70/(1 -...- x^Pell(n)/(1 -...)))))))), a continued fraction.

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A351972 a(n) = 1 + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

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A351973 a(n) = (-1)^n + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1).

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A253190 Triangle T(n, m)=Sum_{k=1..(n-m)/2} C(m+k-1, k)*T((n-m)/2, k), T(n,n)=1.

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A367637 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^6)).

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A367794 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^4)).

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A367800 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^8)).

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A036675 G.f. satisfies A(x) = 1 + xA(x)^2A(x^2).