A027691 -id:A027691 - cp's OEIS frontend

A152950 a(n) = 3 + n*(n-1)/2.

3, 4, 6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81, 94, 108, 123, 139, 156, 174, 193, 213, 234, 256, 279, 303, 328, 354, 381, 409, 438, 468, 499, 531, 564, 598, 633, 669, 706, 744, 783, 823, 864, 906, 949, 993, 1038, 1084, 1131, 1179, 1228, 1278, 1329, 1381, 1434, 1488

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

a(1)=3; then add 1 to the first number, then 2, 3, 4, ... and so on.

Numbers m such that 8*m - 23 is a square. - Bruce J. Nicholson, Jul 25 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000217, A027691, A152947, A152948, A152949.

[3+n*(n-1)/2 : n in [1..50]]; // Wesley Ivan Hurt, Mar 25 2020

A152950:=n->3 + n*(n-1)/2; seq(A152950(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

s=3;lst={3};Do[s+=n;AppendTo[lst,s],{n,1,5!}];lst
Table[3 + n*(n-1)/2, {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n)=3+n*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

[3+binomial(n,2) for n in range(1, 55)] # Zerinvary Lajos, Mar 12 2009

a(n) = A152949(n+1) = 3 + A000217(n-1). - R. J. Mathar, Jan 03 2009
a(n) = 3 + C(n,2), n >= 1. - Zerinvary Lajos, Mar 12 2009
a(n) = a(n-1) + n - 1 (with a(1)=3). - Vincenzo Librandi, Nov 27 2010
Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(23)*Pi/2)/sqrt(23). - Amiram Eldar, Dec 13 2022
From Elmo R. Oliveira, Nov 18 2024: (Start)
G.f.: x*(3 - 5*x + 3*x^2)/(1-x)^3.
E.g.f.: exp(x)*(3 + x^2/2) - 3.
a(n) = A027691(n-1)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A027721 Palindromes of form k^2 + k + 6.

Original entry on oeis.org

6, 8, 606, 656, 818, 83238, 6177716, 6505056, 8343438, 8681868, 834545438, 21543634512, 69872727896, 83456565438, 85425552458, 8032131312308, 8255470745528, 8345676765438, 8716172716178, 8766434346678, 8915858585198, 8973808083798, 213296020692312

Offset: 1

Patrick De Geest

Palindromes h such that 4*h - 23 is a square. - Bruno Berselli, Aug 29 2018

Giovanni Resta, Table of n, a(n) for n = 1..34
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027729, A027691, A027728, A027723.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 6; Select[f@ Range[0, 10^5], palQ] (* Giovanni Resta, Aug 29 2018 *)

a(n) = A027729(n)^2 + A027729(n) + 6. - Giovanni Resta, Aug 27 2018

More terms from Giovanni Resta, Aug 27 2018

A167614 a(n) = (n^2 + 3*n + 8)/2.

Original entry on oeis.org

6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81, 94, 108, 123, 139, 156, 174, 193, 213, 234, 256, 279, 303, 328, 354, 381, 409, 438, 468, 499, 531, 564, 598, 633, 669, 706, 744, 783, 823, 864, 906, 949, 993, 1038, 1084, 1131, 1179, 1228, 1278, 1329, 1381, 1434, 1488, 1543

Offset: 1

Vincenzo Librandi, Nov 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016028, A027691, A152949, A152950, A228446.

[(n^2+3*n+8)/2: n in [1..60]]; // Vincenzo Librandi, Sep 16 2013

Table[(n(n+3))/2+4,{n,80}]  (* Harvey P. Dale, Mar 24 2011 *)
CoefficientList[Series[(6 - 9 x + 4 x^2)/(1 - x)^3,{x, 0, 60}], x] (* Vincenzo Librandi, Sep 16 2013 *)

a(n)=n*(n+3)/2+4 \\ Charles R Greathouse IV, Jan 11 2012

print([n*(n+3)//2+4 for n in range(1, 60)]) # Gennady Eremin, Feb 03 2022

a(n) = n + a(n-1) + 1, with n > 1, a(1)=6.
G.f.: x*(6 - 9*x + 4*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 16 2013
A228446(a(n)) = 7. - Reinhard Zumkeller, Mar 12 2014
a(n) = A152950(n+2) = A152949(n+3) = A016028(n+5). - Mathew Englander, Feb 03 2022
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(4 + 2*x + x^2/2) - 4.
a(n) = A027691(n+1)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)

Corrected (changed one term from 1036 to 1038) by Harvey P. Dale, Mar 24 2011

New name from Charles R Greathouse IV, Jan 11 2012

A027729 Numbers k such that k^2+k+6 is a palindrome.

Original entry on oeis.org

0, 1, 24, 25, 28, 288, 2485, 2550, 2888, 2946, 28888, 146777, 264334, 288888, 292276, 2834101, 2873233, 2888888, 2952316, 2960816, 2985943, 2995631, 14604657, 16353547, 28888888, 29190748, 29585508, 148278137, 264056434, 288888888, 2853889203, 2931604151, 28988127118

Offset: 1

Patrick De Geest

Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027721, A027691, A027718, A027722.

[n: n in [0..3*10^6] | Intseq(n^2+n+6) eq Reverse(Intseq(n^2+n+6))]; // Vincenzo Librandi, Jun 16 2016

palOblongPlus6Q[n_] := Module[{d = IntegerDigits[n^2 + n + 6]}, d == Reverse[d]]; Select[Range[0, 3000000],  palOblongPlus6Q] (* Harvey P. Dale, Nov 14 2012 *)

a(23)-a(33) from Giovanni Resta, Aug 27 2018

A213921 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 clockwise. Table T(n,k) read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 05 2013

A 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) is 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(2,n), T(4,n), ... T(n,4), T(n,2);
...

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   8  14  22  32 ...
   7   9   6  11  18  27 ...
  13  16  12  15  23  33 ...
  21  25  20  24  19  28 ...
  31  36  30  35  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  8,  9, 13;
  17, 14,  6, 16, 21;
  26, 22, 11, 12, 25, 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 W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

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-(j%2)*i+2-int((j+2)/2)
else:
   result=j*j-((i%2)+1)*j + int((i+3)/2)

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

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).

cp's OEIS Frontend

A152950 a(n) = 3 + n*(n-1)/2.

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A027721 Palindromes of form k^2 + k + 6.

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A167614 a(n) = (n^2 + 3*n + 8)/2.

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A027729 Numbers k such that k^2+k+6 is a palindrome.

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A213921 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 clockwise. Table T(n,k) read by antidiagonals.

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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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