A027694 -id:A027694 - cp's OEIS frontend

A176271 The odd numbers as a triangle read by rows.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Reinhard Zumkeller, Apr 13 2010

A108309(n) = number of primes in n-th row.

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   1;
   3,  5;
   7,  9, 11;
  13, 15, 17, 19;
  21, 23, 25, 27, 29;
  31, 33, 35, 37, 39, 41;
  43, 45, 47, 49, 51, 53, 55;
  57, 59, 61, 63, 65, 67, 69, 71;
  73, 75, 77, 79, 81, 83, 85, 87, 89; (End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Nicomachus's Theorem
Wikipedia, Nikomachos von Gerasa

Cf. A000466, A000537, A000578, A001105, A002061, A005408, A014209.
Cf. A016754, A027688, A027690, A027692, A027694, A028387, A048058.
Cf. A053755, A065599, A082111, A108195, A108309, A214604, A214661.

a176271 n k = a176271_tabl !! (n-1) !! (k-1)
a176271_row n = a176271_tabl !! (n-1)
a176271_tabl = f 1 a005408_list where
   f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, May 24 2012

[n^2-n+2*k-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

A176271 := proc(n,k)
    n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013

Table[n^2-n+2*k-1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[n^2-n+2*k-1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.
T(n, k) = A005408(n*(n-1)/2 + k - 1).
T(2*n-1, n) = A016754(n-1) (main diagonal).
T(2*n, n) = A000466(n).
T(2*n, n+1) = A053755(n).
T(n, k) + T(n, n-k+1) = A001105(n), 1 <= k <= n.
T(n, 1)   = A002061(n), central polygonal numbers.
T(n, 2)   = A027688(n-1) for n > 1.
T(n, 3)   = A027690(n-1) for n > 2.
T(n, 4)   = A027692(n-1) for n > 3.
T(n, 5)   = A027694(n-1) for n > 4.
T(n, 6)   = A048058(n-1) for n > 5.
T(n, n-3) = A108195(n-2) for n > 3.
T(n, n-2) = A082111(n-2) for n > 2.
T(n, n-1) = A014209(n-1) for n > 1.
T(n, n) = A028387(n-1).
Sum_{k=1..n} T(n, k) = A000578(n) (Nicomachus's theorem).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).
Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) = A000537(n) (sum of first n rows).

A027726 Numbers k such that k^2+k+9 is a palindrome.

Original entry on oeis.org

0, 1, 9, 11, 13, 22, 30, 31, 138, 300, 304, 305, 331, 438, 969, 1141, 1413, 2367, 3000, 3144, 3881, 9854, 30000, 30605, 72062, 106801, 114141, 125206, 128348, 300000, 315165, 963304, 980560, 989154, 2378507, 3000000, 3040604, 3045679, 3152290, 3932806

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..71
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027727, A027694, A027757, A027758, A027724.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 9; Select[Range[0, 3*10^5], palQ@ f@ # &]

More terms from Giovanni Resta, Aug 29 2018

A027727 Palindromes of form k^2 + k + 9.

Original entry on oeis.org

9, 11, 99, 141, 191, 515, 939, 1001, 19191, 90309, 92729, 93339, 109901, 192291, 939939, 1303031, 1997991, 5605065, 9003009, 9887889, 15066051, 97111179, 900030009, 936696639, 5193003915, 11406560411, 13028282031, 15676667651, 16473337461, 90000300009

Offset: 1

Patrick De Geest

Palindromes h such that 4*h - 35 is a square. - Bruno Berselli, Aug 29 2018

Giovanni Resta, Table of n, a(n) for n = 1..71
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027726, A027694, A027757, A027758, A027725.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 9; Select[f@ Range[0, 10^5], palQ] (* Giovanni Resta, Aug 29 2018 *)
Select[Table[k^2+k+9,{k,0,300000}],PalindromeQ] (* Harvey P. Dale, Nov 21 2024 *)

More terms from Giovanni Resta, Aug 29 2018

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A055630 Table T(k,m) = k^2 + m read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 4, 2, 2, 9, 5, 3, 3, 16, 10, 6, 4, 4, 25, 17, 11, 7, 5, 5, 36, 26, 18, 12, 8, 6, 6, 49, 37, 27, 19, 13, 9, 7, 7, 64, 50, 38, 28, 20, 14, 10, 8, 8, 81, 65, 51, 39, 29, 21, 15, 11, 9, 9, 100, 82, 66, 52, 40, 30, 22, 16, 12, 10, 10, 121, 101, 83, 67, 53, 41, 31, 23, 17, 13

Offset: 0

Henry Bottomley, Jun 05 2000

			Table begins:
..0...1...4...9..16..25..36..49..64..81.100.121.144...
..1...2...5..10..17..26..37..50..65..82.101.122.145...
..2...3...6..11..18..27..38..51..66..83.102.123.146...
..3...4...7..12..19..28..39..52..67..84.103.124.147...
..4...5...8..13..20..29..40..53..68..85.104.125.148...
..5...6...9..14..21..30..41..54..69..86.105.126.149...
..6...7..10..15..22..31..42..55..70..87.106.127.150...
..7...8..11..16..23..32..43..56..71..88.107.128.151...
..8...9..12..17..24..33..44..57..72..89.108.129.152...
..9..10..13..18..25..34..45..58..73..90.109.130.153...
.10..11..14..19..26..35..46..59..74..91.110.131.154...
... - _Philippe Deléham_, Mar 31 2013

First column is A001477, second column is A000027, first row is A000290, second row is A002522, third row (apart from first term) is A010000, main diagonal is A002378, other diagonals include A028387, A028552, A014209, A002061, A014206, A027688-A027694, each row of A055096 (as upper right triangle) is right hand part of some row of this table

Cf. A185910

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A176271 The odd numbers as a triangle read by rows.

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A027726 Numbers k such that k^2+k+9 is a palindrome.

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A027727 Palindromes of form k^2 + k + 9.

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A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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A055630 Table T(k,m) = k^2 + m read by antidiagonals.

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