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A383691 Square numbers with distinct digits from 1-9 that have an initial string of two or more digits forming a square number.

169, 256, 361, 3249, 16384, 18496, 36481, 81796, 237169, 729316, 2537649, 3481956, 5184729, 36517849, 81432576, 254817369, 361874529, 529874361

Offset: 1

Hilarie Orman, May 05 2025

The sequence is based on correspondence with Donald S. McDonald. He originated the idea and computed the complete list of terms.

The last two numbers in the sequence are perfect squares with square roots that are digit permutations of each other: 361874529 = 19023*19023 and 529874361 = 23019*23019.

			169 is a term, because 169 and 16 are both squares, and each digit of 169 is unique.
18496 is a term, because 18496 and 1849 are both squares, and each digit of 18496 is unique.

Cf. A036744, A036745, A352329.

dd=Select[Range[11,25000]^2,IntegerLength[#]==CountDistinct[IntegerDigits[#]]&&ContainsNone[IntegerDigits[#],{0}]&];f[n_]:=AnyTrue[Table[FromDigits[Take[IntegerDigits[n],i]],{i,2,IntegerLength[n]-1}]^(1/2),IntegerQ];Select[dd,f[#]&] (* James C. McMahon, May 07 2025 *)

A340635 Number of free deltoidal hexecontahedron polykites with n faces.

Original entry on oeis.org

1, 2, 4, 10, 27, 80, 238, 751, 2374, 7608, 24371, 78341, 251269, 804594, 2565683, 8140581, 25650546, 80113477, 247328763, 752292342, 2245589193, 6549404314, 18573169874, 50940335705, 134325417269

Offset: 1

Hilarie Orman, Jan 14 2021

The surface of a deltoidal hexecontahedron has 60 congruent faces. Edge-connected faces are called polykites. Reflections are included separately in the counts. Polykites can have from 1 to 60 faces.

These polykites are counted up to the full icosahedral group (order 120) of the deltoidal hexecontahedron. - Peter Kagey, Apr 28 2025

Wikipedia, Deltoidal Hexecontahedron

Cf. A057786.

Cf. A030137 (dodecahedron), A030138 (icosahedron).

a(13)-a(25) from Bert Dobbelaere, Jun 13 2025

A327665 Fibonacci with binary selection.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 11, 20, 34, 56, 121, 244, 481, 938, 1832, 3540, 6757, 12708, 23410, 42328, 73764, 122889, 275122, 408967, 832560, 1580364, 2834653, 5044480, 8652446, 13975133, 29832886, 58354102, 112803422, 215061934, 401711254, 737267883, 1313954863, 2259026414

Offset: 0

Hilarie Orman, Sep 21 2019

Hilarie Orman, C program for generating Fibonacci with binary selection

Cf. A000045, A007088.

e[n_] := Floor[Log2[n]]; a[0] = 0; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[BitGet[a[n - 1], em - i] * a[n - 2 - i], {i, 0, (em = e[a[n - 1]])}]; Array[a, 38, 0] (* Amiram Eldar, Sep 28 2019 *)

lista(nn) = {my(va = vector(nn)); va[1] = 0; va[2] = 1; va[3] = 1; for (n=4, nn, my(b = binary(va[n-1])); va[n] = va[n-1] + sum(k=1, #b, b[k]*va[n-k-1]);); va;} \\ Michel Marcus, Sep 28 2019

def aupton(nn):
  alst = [0, 1, 1]
  for n in range(3, nn+1):
    b = list(map(int, bin(alst[n-1])[2:]))
    alst.append(alst[n-1] + sum(bi*alst[n-i-2] for i, bi in enumerate(b)))
  return alst[:nn+1]
print(aupton(37)) # Michael S. Branicky, Jan 19 2021

a(n) = a(n-1) + Sum_{i=0..e(a(n-1))} b(a(n-1), e(a(n-1))-i)*a(n-i-2) where b(k, i) is the i-th bit in the binary expansion of k, with b(k, 0) being the low order bit of k, and e(k) = floor(log_2(k)). The initial terms are a(0) = 0, a(1) = 1. [edited by Michel Marcus, Sep 28 2019 and Michael S. Branicky, Jan 19 2021]

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User: Hilarie Orman

A383691 Square numbers with distinct digits from 1-9 that have an initial string of two or more digits forming a square number.

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A340635 Number of free deltoidal hexecontahedron polykites with n faces.

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A327665 Fibonacci with binary selection.

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