A367362 -id:A367362 - cp's OEIS frontend

A367610 Comma transform of A367362.

11, 22, 33, 44, 55, 66, 77, 88, 99, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 21, 22, 23, 24, 25, 26, 27, 28, 29, 33, 31, 32, 33, 34, 35, 36, 37, 38, 39, 44, 41, 42, 43, 44, 45, 46, 47, 48, 49, 55, 51, 52, 53, 54, 55, 56, 57, 58, 59, 66, 61, 62, 63, 64, 65, 66, 67, 68, 69, 77, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is the second-order comma transform of the nonnegative integers.

See A367360 for further information.

N. J. A. Sloane, Table of n, a(n) for n = 1..9999

Cf. A367360, A367362.

from itertools import count, islice, pairwise
def S(): yield from (str(i) for i in count(0))
def C(): yield from (str(int(t[-1]+u[0])) for t, u in pairwise(S()))
def a(): yield from (int(t[-1]+u[0]) for t, u in pairwise(C()))
print(list(islice(a(), 80))) # Michael S. Branicky, Dec 11 2023

A367360 Comma transform of squares.

Original entry on oeis.org

1, 14, 49, 91, 62, 53, 64, 96, 48, 11, 1, 11, 41, 91, 62, 52, 62, 93, 43, 14, 4, 14, 45, 95, 66, 56, 67, 97, 48, 19, 9, 11, 41, 91, 61, 51, 61, 91, 41, 11, 1, 11, 41, 91, 62, 52, 62, 92, 42, 12, 2, 12, 42, 92, 63, 53, 63, 93, 43, 13, 3, 13, 43, 94, 64, 54, 64, 94, 44, 14, 5, 15, 45, 95, 65, 55, 65, 96, 46, 16, 6, 16, 46, 97

Offset: 0

N. J. A. Sloane, Nov 22 2023

To compute the comma transform of a sequence [b,c,d,e,f,...], concatenate the last digit of each term with the first digit of the following term. In other words, these are the numbers formed by the pairs of digits that surround the commas that separate the terms of the original sequence.
The comma transform CT(S) of a sequence S of positive numbers maps S into the set F consisting of finite or infinite sequences of positive numbers each with one or two digits. The inverse comma transform CTi maps an element of F to an element of F.
Inspired by Eric Angelini's A121805.

			The squares are 0, 1, 4, 9, 16, 25, ..., so the comma transform is [0]1, 14, 49, 91, 62, ...

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A121805, A008959, A002993.

A166499 is the comma transform of the primes, A367361 of the powers of 2, A367362 of the nonnegative integers. See also A368362.

Maple code for comma transform (CT(a)) of a sequence a:
# leading digit, from A000030
Ldigit:=proc(n) local v; v:=convert(n, base, 10); v[-1]; end;
CT:=proc(a) local b,i; b:=[];
for i from 1 to nops(a)-1 do
b := [op(b), 10*(a[i] mod 10) + Ldigit(a[i+1])]; od: b; end;
# Inverse comma transform of sequence A calculated in base "bas": - N. J. A. Sloane, Jan 03 2024
bas := 10;
Ldigit:=proc(n) local v; v:=convert(n, base, bas); v[-1]; end;
CTi := proc(A) local B,i,L,R;
for i from 1 to nops(A) do
   if A[i]>=bas^2 then error("all terms must have 1 or 2 digits"); fi; od:
B:=Array(1..nops(A),-1);
if A[1] >= bas then B[1]:= Ldigit(A[1]); L:=(A[1] mod bas);
else B[1]:=10; L:=A[1];
fi;
for i from 2 to nops(A) do
  if A[i] >= bas then R := Ldigit(A[i]) else R:=0; fi;
  B[i] := L*bas + R;
  L := (A[i] mod bas);
od;
B;
end;
# second Maple program:
a:= n-> parse(cat(""||(n^2)[-1],""||((n+1)^2)[1])):
seq(a(n), n=0..99);  # Alois P. Heinz, Nov 22 2023

a[n_]:=FromDigits[{Last[IntegerDigits[n^2]],First[IntegerDigits[(n+1)^2]]}];
a/@Range[0,83] (* Ivan N. Ianakiev, Nov 24 2023 *)

from itertools import count, islice, pairwise
def S(): yield from (str(i**2) for i in count(0))
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 84))) # Michael S. Branicky, Nov 22 2023

def A367360(n): return (0, 10, 40, 90, 60, 50, 60, 90, 40, 10)[n%10]+int(str((n+1)**2)[0]) # Chai Wah Wu, Dec 22 2023

a(n) = 10 * A008959(n) + A002993(n+1). - Alois P. Heinz, Nov 22 2023

A368362 Inverse comma transform of 1,2,3,4,5,...,99.

Original entry on oeis.org

10, 10, 20, 30, 40, 50, 60, 70, 80, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 92, 2, 12, 22, 32, 42, 52, 62, 72, 82, 93, 3, 13, 23, 33, 43, 53, 63, 73, 83, 94, 4, 14, 24, 34, 44, 54, 64, 74, 84, 95, 5, 15, 25, 35, 45, 55, 65, 75, 85, 96, 6, 16, 26, 36, 46, 56, 66, 76, 86, 97, 7, 17, 27, 37, 47, 57, 67, 77, 87, 98, 8, 18, 28, 38, 48, 58, 68, 78, 88, 99, 9, 19, 29, 39, 49, 59, 69, 79, 89

Offset: 1

N. J. A. Sloane, Jan 03 2024

See A367360 for further information.

Cf. A367360, A367362

A369303 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number whose string value contains the comma transform (cf. A367360) of the previous two terms.

Original entry on oeis.org

1, 2, 12, 21, 22, 112, 121, 210, 120, 10, 11, 13, 110, 31, 3, 113, 131, 231, 122, 111, 211, 123, 114, 310, 43, 4, 34, 143, 41, 134, 115, 141, 51, 15, 116, 151, 61, 16, 117, 161, 71, 17, 118, 171, 81, 18, 119, 181, 91, 19, 311, 93, 190, 312, 23, 220, 32, 30, 223, 20, 132, 14, 212, 42, 24, 221

Offset: 1

Scott R. Shannon, Jan 19 2024

The sequence is conjectured to be a permutation of the positive integers. The fixed points begin 1, 2, 10, 11, 2863, 3164, 3545, 3947, 6835, 6947, 7052, ... although it is likely there are infinitely more.

			a(3) = 12 as the comma transform of 1 and 2 is 12.
a(6) = 112 as the comma transform of 21 and 22 is 12, but 12 has already appeared so the next lowest unused number to contain '12' is 112.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first one million terms. The green line is a(n) = n.

Cf. A367360, A121805, A367362.

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A367610 Comma transform of A367362.

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A367360 Comma transform of squares.

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A368362 Inverse comma transform of 1,2,3,4,5,...,99.

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A369303 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number whose string value contains the comma transform (cf. A367360) of the previous two terms.

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