A004019 -id:A004019 - cp's OEIS frontend

A059417 Start with 1; square; add 2; subtract 1; repeat.

1, 1, 3, 2, 4, 6, 5, 25, 27, 26, 676, 678, 677, 458329, 458331, 458330, 210066388900, 210066388902, 210066388901, 44127887745906175987801, 44127887745906175987803, 44127887745906175987802

Offset: 0

Jonathan Scharff (jonscharff(AT)home.com), Jan 30 2001

a(22) has 46 digits.

Seen on a quiz.

Harry J. Smith, Table of n, a(n) for n = 0..36

Cf. A004019.

Flatten@ NestList[# + {0, 2, 1} &[Last[#]^2] &, {1}, 7] (* Michael De Vlieger, Sep 06 2016 *)

{ for (n = 0, 36, if (n==0, a=1, if (n%3 == 1, a*=a, if (n%3==2, a+=2, if (n%3==0, a-=1)))); write("b059417.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 26 2009

a(1+3k) = A004019(k+1), a(2+3k) = A004019(k+1)+2, a(3+3k) = A004019(k+1)+1, k>=0. - Yuriy Sibirmovsky, Sep 04 2016

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

A060137 Square array read by antidiagonals with T(n,k)=T(n,k-1)^2-n*T(n,k-1)+1 and T(n,0)=0.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 5, 1, 1, 0, 26, 1, 0, 1, 0, 677, 1, 1, -1, 1, 0, 458330, 1, 0, 5, -2, 1, 0, 210066388901, 1, 1, 11, 13, -3, 1, 0, 44127887745906175987802, 1, 0, 89, 118, 25, -4, 1, 0, 1947270476915296449559703445493848930452791205, 1, 1, 7655, 13453, 501, 41, -5, 1, 0

Offset: 0

Henry Bottomley, Mar 05 2001

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Rows include A003095, A057427, A000035. Columns include A000004, A000012, A022958, A001844 (offset). Cf. A060136.

T(0,k)=A004019(k-1)+1=A056207(k-2)+2. - R. J. Mathar, Apr 24 2007

A067691 a(n) = (a(n-1) + 1)^(n-1) for n > 0, a(0) = 0.

Original entry on oeis.org

0, 1, 2, 9, 1000, 1004006004001, 1020191144865623440455270145683555422808365843606721760320032

Offset: 0

Reinhard Zumkeller, Feb 04 2002

nonn

a(7) has 361 digits.

			Given a(3) = 9: a(4) = (a(4-1)+1)^(4-1) = (9+1)^3 = 1000.

Cf. A004019.

a(n) = if (n==0, 0, (a(n-1) + 1)^(n-1)); \\ Michel Marcus, Feb 05 2021

A100523 a(n) = a(n-1)^2 + 2*a(n-1) - 1 with a(0) = 1.

Original entry on oeis.org

1, 2, 7, 62, 3967, 15745022, 247905749270527, 61457260521381894004129398782, 3776994870793005510047522464634252677140721938309041881087

Offset: 0

Reinhard Zumkeller, Nov 24 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..12
Index entries for sequences of form a(n+1) = a(n)^2 + ...

Cf. A004019.

[n le 1 select 1 else Self(n-1)^2 +2*Self(n-1) -1: n in [1..13]]; // G. C. Greubel, Jun 26 2022

RecurrenceTable[{a[n] == a[n-1]^2 + a[n-1]*2 - 1, a[0] == 1}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
NestList[#^2+2#-1&,1,10] (* Harvey P. Dale, Aug 17 2025 *)

def a(n): return 1 if (n==0) else a(n-1)^2 + 2*a(n-1) - 1 # a=A100523
[a(n) for n in (0..12)] # G. C. Greubel, Jun 26 2022

a(n) ~ c^(2^n), where c = 1.6784589651254290832096890907802189718037513767396728769965837700954845976... . - Vaclav Kotesovec, Dec 18 2014

A157679 Number of subtrees of a complete binary tree.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 15, 25, 35, 49, 70, 100, 160, 256, 416, 676, 936, 1296, 1800, 2500, 3550, 5041, 7171, 10201, 16261, 25921, 41377, 66049, 107169, 173889, 282309, 458329, 634349, 877969, 1215289, 1682209, 2335897, 3243601, 4504301, 6255001, 8881051, 12609601

Offset: 0

Paolo Bonzini, Mar 04 2009

Take the complete binary tree with n labeled nodes. Here is a poor picture of the tree with 6 nodes:
        R
       / \
      /   \
     /     \
    o       o
   / \     /
  o   o   o
Then the number of rooted subtrees of the graph is a(n).

			For n = 3, the a(3) = 4 subtrees are:
  R   R   R      R
     /     \    / \
    o       o  o   o

Alois P. Heinz, Table of n, a(n) for n = 0..5971
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences related to rooted trees

Cf. A004019, A005468.

a:= proc(n) option remember; `if`(n<2, n,
     (h-> (1+a(h))*(1+a(n-1-h)))(iquo(n, 2)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 02 2022

a[0] = 0; a[1] = 1; a[n_?EvenQ] := a[n] = (1 + a[n/2 - 1])*(1 + a[n/2]); a[n_?OddQ] := a[n] = (1 + a[(n-1)/2])^2; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 19 2011 *)

a(0) = 0, a(1) = 1.
a(n) = 1 + a(floor((n-1)/2)) + a(ceiling((n-1)/2)) + a(floor((n-1)/2)) * a(ceiling((n-1)/2)) = (1+a(floor((n-1)/2))) * (1+a(ceiling((n-1)/2))).
If b(n) is sequence A005468, then a(n)=b(n+1)-1. From the definition of A005468, a(n) = b(floor((n+1)/2))*b(ceiling((n+1)/2)). So for every odd n a(n) is a square: a(2*n-1)=b(n)^2.
If c(n) is sequence A004019, then c(n)=a(2^n-1).
A004019 (and Aho and Sloane) give a closed formula for c(n) that translates to a(n) = nearest integer to b^((n+1)/2) - 1" where b = 2.25851...; the formula gives the asymptotic behavior of this sequence, however it does not compute the correct values for a(n) unless n+1 is a power of two.

A337510 a(n) = Sum_{k=0..n} T(n,k) where T(n,k) = (T(n-1, k-1) + T(n-1,k))^2.

Original entry on oeis.org

1, 2, 6, 52, 3854, 21090612, 629815387162156, 561871511512925116799625359336, 446575758106416254441837050759254156476271759098752411181598

Offset: 0

Glen Gilchrist, Aug 30 2020

Based on Pascal's triangle A007318 by additionally squaring the sum of each term generated. For example, in Pascal, n=3 gives 1,2,1. Here n=3 gives, 1^2, (1+1)^2, 1^2 = 1+4+1.

			1 = 1
1 + 1 = 2
1 + (1 + 1)^2  + 1 = 1 + 4 + 1 = 6
1 + (1 + 4)^2  + (4 + 1)^2 + 1 = 1 + 25 + 25 + 1 = 52
1 + (1 + 25)^2 + (25 + 25)^2 + (25 + 1)^2 + 1 = 1 + 676 + 2500 + 676 + 1 = 3854.

Cf. A004019, A327563, A007318.

def r(i):
  t = [[0, 1, 0], [0, 1, 1, 0]]
  for n in range(2, i+1):
    t.append([0])
    for k in range(1, n+2):
      t[n].append((t[n-1][k-1] + t[n-1][k])**2)
    t[n].append(0)
  return(sum(t[i]))

a(n) = Sum_{k=0..n} T(n,k) where T(n,k) = (T(n-1,k-1) + T(n-1,k))^2; T(0,0)=1; T(n,-1):=0; T(n,k):=0, n < k.

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A059417 Start with 1; square; add 2; subtract 1; repeat.

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A060137 Square array read by antidiagonals with T(n,k)=T(n,k-1)^2-n*T(n,k-1)+1 and T(n,0)=0.

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A067691 a(n) = (a(n-1) + 1)^(n-1) for n > 0, a(0) = 0.

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A100523 a(n) = a(n-1)^2 + 2*a(n-1) - 1 with a(0) = 1.

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A157679 Number of subtrees of a complete binary tree.

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A337510 a(n) = Sum_{k=0..n} T(n,k) where T(n,k) = (T(n-1, k-1) + T(n-1,k))^2.

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