A017317 -id:A017317 - cp's OEIS frontend

A168420 a(n) = 4 + 10*floor(n/2).

4, 14, 14, 24, 24, 34, 34, 44, 44, 54, 54, 64, 64, 74, 74, 84, 84, 94, 94, 104, 104, 114, 114, 124, 124, 134, 134, 144, 144, 154, 154, 164, 164, 174, 174, 184, 184, 194, 194, 204, 204, 214, 214, 224, 224, 234, 234, 244, 244, 254, 254, 264, 264, 274, 274, 284

Offset: 1

Vincenzo Librandi, Nov 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017317, A168278.

[4+10*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

Table[4 + 10 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[(4 + 10 x - 4 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)

a(n) = 10*n - a(n-1) - 2, with n>1, a(1)=4.
G.f.: 2*x*(2 + 5*x - 2*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 19 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
a(n) = 2*A168278(n). - Bruno Berselli, Sep 19 2013
E.g.f.: (1/2)*(5 - 8*exp(x) + (10*x + 3)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 22 2016

New definition by Vincenzo Librandi, Sep 19 2013

A266297 Numbers whose last digit is a square.

Original entry on oeis.org

0, 1, 4, 9, 10, 11, 14, 19, 20, 21, 24, 29, 30, 31, 34, 39, 40, 41, 44, 49, 50, 51, 54, 59, 60, 61, 64, 69, 70, 71, 74, 79, 80, 81, 84, 89, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 120, 121, 124, 129, 130, 131, 134, 139, 140, 141, 144, 149

Offset: 1

Wesley Ivan Hurt, Dec 26 2015

Numbers ending in 0, 1, 4 and 9.
Union of A008592, A017281, A017317 and A017377. - Hurt
None of these numbers are prime in Z[phi] (where phi = 1/2 + sqrt(5)/2 is the golden ratio), since the numbers in this sequence that are prime in Z can be expressed in the form (a - b sqrt(5))(a + b sqrt(5)). - Alonso del Arte, Dec 30 2015
Union of A197652 and A016897. - Wesley Ivan Hurt, Dec 31 2015
Union of A146763 and A090771. - Wesley Ivan Hurt, Jan 01 2016

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008592, A016897, A017281, A017317, A017377, A090771, A091084, A141158, A146763, A197652.

[(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n) div 4))/4: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A266297:=n->(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/4: seq(A266297(n), n=1..100);

Table[(10 n - 11 + (-1)^n + (4 + 2 (-1)^n)*(-1)^((2 n - 1 + (-1)^n)/4))/4, {n, 50}] (* G. C. Greubel, Dec 27 2015 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 4, 9, 10}, 60] (* Vincenzo Librandi, Dec 27 2015 *)
CoefficientList[Series[x*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Dec 30 2015 *)
Flatten[Table[10n + {0, 1, 4, 9}, {n, 0, 19}]] (* Alonso del Arte, Dec 30 2015 *)
Select[Range[0,150],MemberQ[{0,1,4,9},Mod[#,10]]&] (* Harvey P. Dale, Jul 30 2019 *)

is(n) = issquare(n%10); \\ Altug Alkan, Dec 29 2015

G.f.: x^2*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)).
a(n) = a(n - 1) + (n - 4) - a(n - 5) for n > 5.
a(n) = (10n - 11 + (-1)^n + (4 + 2(-1)^n) * (-1)^((2n - 1 + (-1)^n)/4))/4.
a(n+1) - a(n) = A091084(n+1) for n>0.
Sum_{n>=2} (-1)^n/a(n) = (14*sqrt(5)*arccoth(sqrt(5)) - 2*Pi*sqrt(1-2/sqrt(5)) + 16*log(2) + 5*log(5))/40. - Amiram Eldar, Jul 30 2024

A271508 Numbers that are congruent to {1,4} mod 10.

Original entry on oeis.org

1, 4, 11, 14, 21, 24, 31, 34, 41, 44, 51, 54, 61, 64, 71, 74, 81, 84, 91, 94, 101, 104, 111, 114, 121, 124, 131, 134, 141, 144, 151, 154, 161, 164, 171, 174, 181, 184, 191, 194, 201, 204, 211, 214, 221, 224, 231, 234, 241, 244, 251, 254, 261, 264, 271, 274

Offset: 1

Wesley Ivan Hurt, Apr 08 2016

Numbers ending in 1 or 4, Union of A017281 and A017317.

a(n+3) gives the sum of 5 consecutive terms of A004442 starting at A004442(n) for n>0. (i.e., a(4) = 14 = 0+3+2+5+4 = Sum_{i=0..4} A004442(n+i)).

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A004442, A017281, A017317, A042948, A047241.

Magma
```
[5*n-5-(-1)^n : n in [1..100]];
```

A271508:=n->5*n-5-(-1)^n: seq(A271508(n), n=1..100);

Table[5 n - 5 - (-1)^n, {n, 60}] (* or *)
Select[Range[0, 300], MemberQ[{1, 4}, Mod[#, 10]] &]

my(x='x+O('x^99)); Vec(x*(1+3*x+6*x^2)/((-1+x)^2*(1+x))) \\ Altug Alkan, Apr 09 2016

G.f.: x*(1+3*x+6*x^2)/((-1+x)^2*(1+x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 5*n - 5 - (-1)^n.
a(n) = -n + 2*A047241(n).
a(n+1) = n + 1 + 2*A042948(n).
Shifted bisections: a(2n+2) = A017317(n), a(2n+1) = A017281(n).
E.g.f.: 5*(x-1)*exp(x) - exp(-x). - G. C. Greubel, Apr 08 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A348548 Positive integers that are the product of two integers ending with 8.

Original entry on oeis.org

64, 144, 224, 304, 324, 384, 464, 504, 544, 624, 684, 704, 784, 864, 944, 1024, 1044, 1064, 1104, 1184, 1224, 1264, 1344, 1404, 1424, 1444, 1504, 1584, 1624, 1664, 1744, 1764, 1824, 1904, 1944, 1984, 2064, 2124, 2144, 2184, 2204, 2224, 2304, 2384, 2464, 2484, 2544

Offset: 1

Stefano Spezia, Oct 22 2021

			64 = 8*8, 144 = 8*18, 224 = 8*28, 304 = 8*38, 324 = 18*18, 384 = 8*48, ...

Cf. A017317 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347253 (ending with 4), A348054 (ending with 7), A348549.

a={}; For[n=0, n<=260, n++, For[k=0, k<=n, k++, If[Mod[10*n+4, 10*k+8]==0 && Mod[(10*n+4)/(10*k+8), 10]==8 && 10*n+4>Max[a], AppendTo[a, 10*n+4]]]]; a

def aupto(lim): return sorted(set(a*b for a in range(8, lim//8+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2550)) # Michael S. Branicky, Oct 22 2021

Lim_{n->infinity} a(n)/a(n-1) = 1.

A348831 Positive numbers whose square starts and ends with exactly 44, and no 444.

Original entry on oeis.org

212, 2112, 6638, 6662, 6688, 20988, 21012, 21062, 21112, 21138, 21162, 21188, 21212, 66338, 66362, 66388, 66412, 66438, 66488, 66512, 66562, 66588, 66612, 66712, 66738, 66762, 66788, 66812, 66838, 66862, 66888, 66912, 66938, 66988, 67012, 67062, 209762, 209788

Offset: 1

Bernard Schott, Nov 08 2021

When a square starts and ends with digits dd, then dd is necessarily 44.
The last 2 digits of terms are either 12, 38, 62 or 88.
From Marius A. Burtea, Nov 09 2021 : (Start)
The sequence is infinite because the numbers 212, 2112, 21112, ..., (19*10^k + 8) / 9, k >= 3, are terms because the remainder when dividing by 1000 is 544 and 445*10^(2*k - 2) < ((19*10^k + 8) / 9)^2 < 447*10^(2*k - 2), k >= 3.
Also 6638, 66338, 663338, 6633338, 66333338, 663333338, 6633333338, ..., (199*10^k + 14) / 3, k >= 2, are terms and have no digits 0, because their squares are: 44063044, 4400730244, 4400730244, 440017302244, 44001173022244, 4400111730222244, 440011117302222244, ... (End)

			212 is a term since 212^2 = 44944.
662 is not a term since 662^2 = 438244.
668 is not a term since 668^2 = 446224.
2108 is not a term since 2108^2 = 4443664.
21038 is not a term since 21038^2 = 442597444.
21088 is not a term since 21088^2 = 444703744.

Cf. A017317.
Subsequence of A045858, A273375, A305719 and A346774.
Similar to: A348488 (d=4), this sequence (dd=44), A348832 (ddd=444).

fd:=func; fs:=func; [n:n in [1..210000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 08 2021

Select[Range[10, 300000], (d = IntegerDigits[#^2])[[1 ;; 2]] ==  d[[-2 ;; -1]] == {4, 4} && d[[-3]] != 4 && d[[3]] != 4 &] (* Amiram Eldar, Nov 08 2021 *)

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-2
def aupto(N):
  ends = [12, 38, 62, 88]
  r = takewhile(lambda x: x<=N, (100*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(209788)) # Michael S. Branicky, Nov 08 2021

cp's OEIS Frontend

A168420 a(n) = 4 + 10*floor(n/2).

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A266297 Numbers whose last digit is a square.

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A271508 Numbers that are congruent to {1,4} mod 10.

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A348548 Positive integers that are the product of two integers ending with 8.

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A348831 Positive numbers whose square starts and ends with exactly 44, and no 444.

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