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A350302 Irregular triangle read by rows: The n-th row lists the larger number in each "n-th power friendship", that is, numbers k such that digsum(digsum(k^n)^n) = k and digsum(k^n) < k.

16, 28, 27, 29, 34, 47, 55, 62, 65, 73, 80, 127, 70, 121, 136, 181, 117, 130, 153, 153, 160, 161, 176, 181, 184, 187, 189, 198, 208, 221, 235, 193, 252, 190, 233, 220, 247, 220, 254, 257, 259, 277, 279, 289, 263, 319, 261, 331, 260, 297, 316, 280, 304, 313

Offset: 2

Daniel Carter, Dec 23 2021

			Triangle begins:
  16;
  28;
  27;
  29, 34;
  ;
  47, 55, 62, 65;
  73;
  80;
  ;
  ;
  127;
  70, 121, 136, 181;
  117, 130;
  153;
  153, 160;
  ...

Cf. A007953, A350300, A350301.

A350301 Irregular triangle read by rows: The n-th row lists the smaller number in each "n-th power friendship", that is, numbers k such that digsum(digsum(k^n)^n) = k and digsum(k^n) > k.

Original entry on oeis.org

13, 19, 18, 23, 31, 38, 44, 46, 56, 64, 35, 109, 52, 112, 118, 127, 97, 108, 144, 88, 144, 139, 152, 153, 154, 173, 178, 189, 172, 199, 203, 187, 234, 127, 188, 148, 238, 115, 229, 238, 239, 245, 234, 244, 245, 283, 252, 283, 161, 271, 288, 163, 259, 295, 316

Offset: 2

Daniel Carter, Dec 23 2021

			Triangle begins:
  13;
  19;
  18;
  23, 31;
  ;
  38, 44, 46, 56;
  64;
  35;
  ;
  ;
  109;
  52, 112, 118, 127;
  97, 108;
  144;
  88, 144;
  ...

Cf. A007953, A350300, A350302.

A350300 Irregular triangle read by rows: The n-th row lists the "n-th power friends", numbers k such that digsum(digsum(k^n)^n) = k but digsum(k^n) is not k.

Original entry on oeis.org

13, 16, 19, 28, 18, 27, 23, 29, 31, 34, 38, 44, 46, 47, 55, 56, 62, 65, 64, 73, 35, 80, 109, 127, 52, 70, 112, 118, 121, 127, 136, 181, 97, 108, 117, 130, 144, 153, 88, 144, 153, 160, 139, 152, 153, 154, 161, 173, 176, 178, 181, 184, 187, 189, 189, 198

Offset: 2

Daniel Carter, Dec 23 2021

Two numbers x and y with x < y are said to be "n-th power friends" if digsum(x^n)=y and digsum(y^n)=x. This sequence lists both x and y; A350301 lists just the x's and A350302 lists just the y's.
The name "n-th power friends" comes from "The Man Who Counted", where the n=2 case is discussed:
"The digits of the number 256 add up to 13. The square of 13 is 169. The digits of 169 add up to 16. As a result, the numbers 13 and 16 have a curious relation, which we could call a quadratic friendship" (p. 33).
There are finitely many such k for a particular n since digsum(digsum(k^n)^n) <= 9n log_10(9n log_10(k)). There are no such k for n=1 since either k is a single digit or else digsum(k) < k.

			Triangle begins:
  13, 16;
  19, 28;
  18, 27;
  23, 29, 31, 34;
  ;
  38, 44, 46, 47, 55, 56, 62, 65;
  64, 73;
  35, 80;
  ;
  ;
  109, 127;
  52, 70, 112, 118, 121, 127, 136, 181;
  97, 108, 117, 130;
  144, 153;
  88, 144, 153, 160;
  ...
18 and 27 are in row n=4 since 18^4 = 104976 and 1 + 0 + 4 + 9 + 7 + 6 = 27, and 27^4 = 531441 and 5 + 3 + 1 + 4 + 4 + 1 = 18.

M. Tahan, The Man Who Counted: A Collection of Mathematical Adventures, W. W. Norton & Company, 1993.

Cf. A007953, A350301, A350302.

from math import log
n = 1
while n <= 50:
    k = 2
    while 9*n*log(9*n*log(k,10),10) >= k:
        s1 = sum(int(d) for d in str(k**n))
        s2 = sum(int(d) for d in str(s1**n))
        if k != s1 and k == s2:
            print(k)
        k += 1
    n += 1

A344363 Decimal expansion of (5^(1/4) + 5^(3/4))/2.

Original entry on oeis.org

2, 4, 1, 9, 5, 2, 5, 1, 5, 3, 0, 5, 1, 6, 6, 5, 3, 3, 0, 9, 6, 4, 0, 3, 2, 1, 8, 0, 2, 1, 7, 0, 7, 6, 5, 3, 8, 6, 5, 1, 8, 1, 7, 8, 5, 7, 9, 3, 8, 5, 4, 7, 0, 8, 4, 6, 8, 3, 2, 5, 5, 3, 8, 2, 8, 9, 5, 8, 8, 4, 0, 4, 2, 5, 3, 9, 8, 9, 9, 6, 8, 5, 7, 3, 5, 8, 0, 1, 5, 5, 0, 8, 2, 4, 1, 8, 4, 6, 7, 8, 4, 8, 7, 3, 8

Offset: 1

Daniel Carter, May 15 2021

Solution for z in the system {x = 1/y + 1/z, y = x + 1/z, z = y + 1/x}. The corresponding values of x and y are (5^(1/4) + 5^(-1/4))/2 and 5^(1/4).

The largest aspect ratio of a set of three rectangles which have the property that any two of them can be scaled, rotated, and joined at an edge to obtain a rectangle with the third aspect ratio. The other two aspect ratios are given in the comment above.

			2.419525153051665330964032180217076538651...

I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq (8.8)

Cf. A011003, A344362.

RealDigits[(Surd[5,4]+Surd[5^3,4])/2,10,120][[1]] (* Harvey P. Dale, Jan 01 2023 *)

my(c=250+150*quadgen(20)); a_vector(len) = digits(sqrtint(floor(c*100^(len-2)))); \\ Kevin Ryde, May 28 2021

Equals sqrt(A090550).

Equals Gamma(1/20)*Gamma(9/20)/(Gamma(3/20)*Gamma(7/20)). [Zucker] - R. J. Mathar, Jun 24 2024

A344362 Decimal expansion of (5^(1/4) + 5^(-1/4))/2.

Original entry on oeis.org

1, 0, 8, 2, 0, 4, 4, 5, 4, 3, 0, 9, 8, 8, 2, 1, 2, 8, 2, 9, 5, 7, 5, 6, 6, 0, 3, 3, 6, 9, 9, 7, 8, 0, 6, 6, 5, 8, 7, 5, 7, 4, 7, 4, 7, 4, 6, 3, 3, 5, 9, 1, 9, 5, 5, 1, 4, 3, 2, 8, 8, 4, 7, 6, 5, 9, 8, 3, 4, 5, 3, 9, 5, 2, 9, 7, 1, 7, 8, 7, 4, 2, 2, 8, 6, 6, 1, 0, 0, 8, 0, 4, 4, 7, 8, 3, 2, 3, 2, 5, 9, 3, 0, 3, 4

Offset: 1

Daniel Carter, May 15 2021

Solution for x in the system {x = 1/y + 1/z, y = x + 1/z, z = y + 1/x}. The corresponding values of y and z are 5^(1/4) and (5^(1/4) + 5^(3/4))/2.

The smallest aspect ratio of a set of three rectangles which have the property that any two of them can be scaled, rotated, and joined at an edge to obtain a rectangle with the third aspect ratio. The other two aspect ratios are given in the comment above.

			1.082044543098821282957566033699780665875...

Cf. A011003, A344363, A293409, A176015.

RealDigits[Cosh[Log[5]/4], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

solve(x=1, 2, 5*x^4 - 5*x^2 - 1) \\ Hugo Pfoertner, May 16 2021

my(c=50+30*quadgen(20)); a_vector(len) = digits(sqrtint(floor(c*100^(len-2)))); \\ Kevin Ryde, May 28 2021

Equals 5*A293409.
Equals sqrt(A176015).
Equals cosh(log(5)/4). - Vaclav Kotesovec, May 28 2021

A340540 Number of walks of length 4n in the first octant using steps (1,1,1), (-1,0,0), (0,-1,0), and (0,0,-1) that start and end at the origin.

Original entry on oeis.org

1, 6, 288, 24444, 2738592, 361998432, 53414223552, 8525232846072, 1443209364298944, 255769050813120576, 47020653859202576640, 8907614785269428079168, 1730208409741026141405696, 343266632435192859791576064, 69350551439109880798294334208

Offset: 0

Daniel Carter, Jan 10 2021

There are no such walks with length that is not a multiple of 4.
a(n) is also the number of arrangements of n copies each of "a", "b", "c", and "d" such that no prefix has more b's, c's, or d's than a's.
The analogous problem in dimensions 1 and 2 are given respectively by A000108 (the Catalan numbers) and A006335.
No closed form is known. In fact, it is not known whether this sequence is D-finite (see Bacher et al.).

Alois P. Heinz, Table of n, a(n) for n = 0..150
Axel Bacher, Manuel Kauers, and Rika Yatchak, Continued Classification of 3D Lattice Walks in the Positive Octant, arXiv:1511.05763 [math.CO], 2015.

Cf. A000108, A006335, A149424.

Column k=3 of A340591.

b:= proc(n, l) option remember; `if`(n=0, 1, `if`(add(i, i=l)+3 x+1, l)), 0) +add(`if`(l[i]>0,
      b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..3))
    end:
a:= n-> b(4*n, [0$3]):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 12 2021

b[n_, l_] := b[n, l] = If[n == 0, 1, If[Total[l] + 3 < n,
  b[n-1, l+1]], 0] + Sum[If[l[[i]] > 0,
  b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, 3}] /. Null -> 0;
a[n_] := b[4n, {0, 0, 0}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 10 2021, after Alois P. Heinz *)

import itertools as it
i = 0
while 1:
    counts = {(a,b,c):0 for a,b,c in it.product(range(i+1), repeat=3)}
    counts[0,0,0] = 1
    for _ in range(4*i):
        update = {(a,b,c):0 for a,b,c in it.product(range(i+1), repeat=3)}
        for x,y,z in counts:
            if counts[x,y,z] != 0:
                for coord in [(x+1,y+1,z+1), (x-1,y,z), (x,y-1,z), (x,y,z-1)]:
                    if coord in update:
                        update[coord] += counts[x,y,z]
        counts = update
    print(i, counts[0,0,0])
    i += 1

cp's OEIS Frontend

User: Daniel Carter

A350302 Irregular triangle read by rows: The n-th row lists the larger number in each "n-th power friendship", that is, numbers k such that digsum(digsum(k^n)^n) = k and digsum(k^n) < k.

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A350301 Irregular triangle read by rows: The n-th row lists the smaller number in each "n-th power friendship", that is, numbers k such that digsum(digsum(k^n)^n) = k and digsum(k^n) > k.

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A350300 Irregular triangle read by rows: The n-th row lists the "n-th power friends", numbers k such that digsum(digsum(k^n)^n) = k but digsum(k^n) is not k.

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A344363 Decimal expansion of (5^(1/4) + 5^(3/4))/2.

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Mathematica

PARI

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A344362 Decimal expansion of (5^(1/4) + 5^(-1/4))/2.

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Mathematica

PARI

PARI

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A340540 Number of walks of length 4n in the first octant using steps (1,1,1), (-1,0,0), (0,-1,0), and (0,0,-1) that start and end at the origin.

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Maple

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Python