A347754 -id:A347754 - cp's OEIS frontend

A347594 a(0) = 1; for n>0, a(n) is the smallest positive integer such that a(n-1)^2 + n^2 + a(n) is a square.

1, 2, 1, 6, 12, 27, 19, 31, 64, 48, 96, 72, 1, 26, 28, 15, 3, 26, 24, 24, 48, 64, 44, 35, 48, 96, 108, 151, 131, 223, 447, 831, 639, 190, 380, 299, 507, 663, 1212, 904, 209, 7, 36, 104, 17, 87, 116, 211, 264, 264, 165, 103, 143, 151, 204, 303, 536, 1055, 860, 1688, 3156, 2592, 1341, 1399

Offset: 0

Scott R. Shannon, Sep 08 2021

In the first one million terms the largest value is a(987016) = 123592518669. In this range the smallest number that has not yet appeared is 9.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 1 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 1 = 9 = 3^2 is the next smallest square.
a(60) = 3156 as a(59)^2 + 60^2 = 2849344 + 3600 = 2852944, and 2852944 + 3156 = 2856100 = 1690^2 is the next smallest square.

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

Cf. A000290, A103605, A009000, A005408, A347754.

Nest[Append[#, Block[{k = 1, m = Last[#1]}, While[! IntegerQ@ Sqrt[#2^2 + m^2 + k], k++]; k]] & @@ {#, Length@ #} &, {1}, 63] (* Michael De Vlieger, Sep 08 2021 *)

lista(nn) = {my(prec = 1, list=List()); listput(list, prec); for (n=1, nn, my(k = 1); while (!issquare(prec^2+n^2+k), k++); listput(list, k); prec = k;); Vec(list);} \\ Michel Marcus, Sep 13 2021

from math import isqrt
A347594_list = [1]
for n in range(1,10**3):
    m = A347594_list[n-1]**2+n**2
    A347594_list.append((isqrt(m)+1)**2-m) # Chai Wah Wu, Sep 12 2021

A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

Original entry on oeis.org

1, 2, 8, 27, 39, 54, 73, 98, 133, 186, 273, 426, 709, 1250, 2305, 4386, 8517, 16746, 33169, 65978, 131557, 262674, 524865, 1049202, 2097829, 4195034, 8389393, 16778058, 33555333, 67109826, 134218753, 268436546, 536872069, 1073743050, 2147484945, 4294968666, 8589936037, 17179870706

Offset: 0

Scott R. Shannon, Sep 08 2021

nonn

This sequence uses the same rules as A347594 except here all numbers must be unique. Up to 10^5 terms all terms are larger than the previous term; it is unknown if this holds for all terms as n->infinity.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 8 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 8 = 16 = 4^2. Note that although 8 + 1 = 9 = 3^2, 1 cannot be chosen as a(0) = 1.
a(3) = 27 as a(2)^2 + 3^2 = 64 + 9 = 73 and 73 + 27 = 100 = 10^2.  Note that although 73 + 8 = 81 = 9^2, 8 cannot be chosen as a(2) = 8.
a(4) = 39 as a(3)^2 + 4^2 = 729 + 16 = 745, and 745 + 39 = 784 = 28^2 is the next smallest square.

Cf. A347594, A000290, A103605, A009000, A005408, A347754.

A349119 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that |n - a(n-1)| + a(n) is a square.

Original entry on oeis.org

1, 4, 2, 3, 8, 6, 9, 7, 15, 10, 16, 11, 24, 5, 27, 13, 22, 20, 14, 31, 25, 12, 26, 33, 40, 21, 44, 19, 55, 23, 18, 36, 32, 35, 48, 51, 34, 46, 17, 42, 47, 30, 37, 43, 63, 82, 28, 45, 61, 52, 62, 38, 50, 78, 57, 79, 41, 65, 29, 70, 39, 59, 97, 66, 98, 67, 80, 68, 49, 101, 69, 119, 53, 124, 71, 60

Offset: 0

Scott R. Shannon, Nov 08 2021

			a(1) = 4 as |1 - a(0)| = |1 - 1| = 0, and 0 + 4 = 4 = 2^2 is the next smallest square. Note a(1) cannot be 1 as a(0) = 1.
a(4) = 8 as |4 - a(3)| = |4 - 3| = 1, and 1 + 8 = 9 = 3^2 is the next smallest square. Note a(4) cannot be 3 as a(3) = 3.
a(8) = 15 as |8 - a(7)| = |8 - 7| = 1, and 1 + 15 = 16 = 4^2 is the next smallest square. Note a(8) cannot be 3 or 8 as these have previously occurred.

Scott R. Shannon, Image of the first 1000000 terms.

Cf. A349182, A000290, A347595, A347594, A009000, A347754.

A349182 a(0) = 1; for n>0, a(n) is the smallest positive integer such that |n - a(n-1)| + a(n) is a square.

Original entry on oeis.org

1, 1, 3, 1, 1, 5, 3, 5, 1, 1, 7, 5, 2, 5, 7, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 11, 9, 6, 2, 8, 2, 6, 9, 11, 1, 1, 13, 11, 8, 4, 12, 6, 12, 4, 8, 11, 13, 1, 1, 15, 13, 10, 6, 1, 10, 3, 10, 1, 6, 10, 13, 15, 1, 1, 17, 15, 12, 8, 3, 14, 7, 16, 7, 14, 3, 8, 12, 15, 17, 1, 1, 19, 17, 14, 10, 5, 18, 11

Offset: 0

Scott R. Shannon, Nov 09 2021

nonn

			a(1) = 1 as |1 - a(0)| = |1 - 1| = 0, and 0 + 1 = 1 = 1^2 is the next smallest square.
a(2) = 3 as |2 - a(1)| = |2 - 1| = 1, and 1 + 3 = 4 = 2^2 is the next smallest square.
a(5) = 5 as |5 - a(4)| = |5 - 1| = 4, and 4 + 5 = 9 = 3^2 is the next smallest square.

Scott R. Shannon, Image for the first 50000 terms.

Cf. A349119, A000290, A347595, A347594, A009000, A347754.

nxt[{n_,a_}]:={n+1,Module[{k=1},While[!IntegerQ[Sqrt[(Abs[n+1-a])+k]],k++];k]}; NestList[ nxt,{0,1},90][[;;,2]] (* Harvey P. Dale, Aug 20 2024 *)

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A347594 a(0) = 1; for n>0, a(n) is the smallest positive integer such that a(n-1)^2 + n^2 + a(n) is a square.

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A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

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A349119 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that |n - a(n-1)| + a(n) is a square.

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A349182 a(0) = 1; for n>0, a(n) is the smallest positive integer such that |n - a(n-1)| + a(n) is a square.

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