A186349 -id:A186349 - cp's OEIS frontend

A186348 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186349.

3, 6, 7, 9, 11, 12, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88

Offset: 1

Clark Kimberling, Feb 20 2011

			First, write
....8....16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36......49....64.......81 (squares)
Then replace each number by its rank, where ties are settled by ranking 8i after the square:
p=(3,6,7,9,11,12,14,16,17,..)=A186348=a(n).
q=(1,2,4,5,8,10,13,15,19,...)=A186349=n+floor((n^2-1)/8).

Cf. A186346, A186347, A186349.

(* adjusted joint rank sequences p and q, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
d=-1/2; u=8; v=0; x=1; y=0;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]];
Table[a[n],{n,1,120}]  (* A186348 *)

a(n)=n+sqrtint(8*n) \\ Charles R Greathouse IV, Jul 05 2013

a(n) = n+floor(sqrt(8n)).

A262221 a(n) = 25n(n + 1)/2 + 1.

Original entry on oeis.org

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651

Offset: 0

Bruno Berselli, Sep 15 2015

Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379.
Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
Cf. centered polygonal numbers listed in A069190.
Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

Magma
```
[25*n*(n+1)/2+1: n in [0..50]];
```

Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* Harvey P. Dale, Jan 29 2023 *)

PARI
```
vector(50, n, n--; 25*n*(n+1)/2+1)
```
Sage
```
[25*n*(n+1)/2+1 for n in (0..50)]
```

G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A123296(n) + 1.
a(n) = A000217(5*n+2) - 2.
a(n) = A034856(5*n+1).
a(n) = A186349(10*n+1).
a(n) = A054254(5*n+2) with n>0, a(0)=1.
a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A186347 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186346.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 26, 30, 34, 38, 43, 48, 53, 58, 64, 70, 76, 82, 89, 96, 103, 110, 118, 126, 134, 142, 151, 160, 169, 178, 188, 198, 208, 218, 229, 240, 251, 262, 274, 286, 298, 310, 323, 336, 349, 362, 376, 390, 404, 418, 433, 448, 463, 478, 494, 510, 526, 542, 559, 576, 593, 610, 628, 646, 664, 682, 701, 720, 739, 758, 778, 798, 818, 838, 859, 880, 901, 922, 944, 966, 988, 1010

Offset: 1

Clark Kimberling, Feb 20 2011

a(n) = a(-8-n) for all n in Z using the formula. - Michael Somos, Apr 05 2024

			First, write
....8....16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36......49....64.......81 (squares)
Then replace each number by its rank, where ties are settled by ranking 8i before the square:
a=(3,5,7,9,11,12,14,15,17,..)=A186346
b=(1,2,4,6,8,10,13,16,19,...)=A186347.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A186346, A186348, A186349.

Mathematica
```
(* See A186346. *)
```

a(n) = (n + 4)^2\8 - 2; \\ Michael Somos, Apr 05 2024

a(n)=n^2\8 + n \\ Charles R Greathouse IV, Apr 11 2024

a(n)=n+floor(sqrt(8n-1/2))=A186346(n).
b(n)=n+floor((n^2+1/2)/8)=A186347(n).
G.f.: x*(1 + x^2 - x^4)/((1 - x)^2 * (1 - x^4)). - Michael Somos, Apr 05 2024

A283394 a(n) = 3n(3*n + 7)/2 + 4.

Original entry on oeis.org

4, 19, 43, 76, 118, 169, 229, 298, 376, 463, 559, 664, 778, 901, 1033, 1174, 1324, 1483, 1651, 1828, 2014, 2209, 2413, 2626, 2848, 3079, 3319, 3568, 3826, 4093, 4369, 4654, 4948, 5251, 5563, 5884, 6214, 6553, 6901, 7258, 7624, 7999, 8383, 8776, 9178, 9589, 10009

Offset: 0

Bruno Berselli, Mar 23 2017

Sum_{k = 0..n} (3*k + r)^3 is divisible by 3*n*(3*n + 2*r + 3)/2 + r^2: the sequence corresponds to the case r = 2 of this formula (other cases are listed in Crossrefs section).
Also, Sum_{k = 0..n} (3*k + 2)^3 / a(n) gives 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, ... (A005449).
a(n) is even if n belongs to A014601. No term is divisible by 3, 5, 7 and 11.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Nickolas Arustamyan, Christopher Cox, Erik Lundberg, Sean Perry, and Zvi Rosen, On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force, arXiv:2106.11416 [math-ph], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005449, A034856, A095794, A101881, A165157, A186349.

Sequences with formula 3*n*(3*n + 2*r + 3)/2 + r^2: A038764 (r=-1), A027468 (r=0), A081271 (r=1), this sequence (r=2), A027468 (r=3; offset: -1), A080855 (r=4; offset: -2).

Magma
```
[3*n*(3*n+7)/2+4: n in [0..50]];
```

Table[3 n (3 n + 7)/2 + 4, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{4,19,43},50] (* Harvey P. Dale, Mar 02 2019 *)

Maxima
```
makelist(3*n*(3*n+7)/2+4, n, 0, 50);
```

a(n) = 3*n*(3*n + 7)/2 + 4; \\ Indranil Ghosh, Mar 24 2017

Python
```
[3*n*(3*n+7)/2+4 for n in range(50)]
```
Sage
```
[3*n*(3*n+7)/2+4 for n in range(50)]
```

O.g.f.: (4 + 7*x - 2*x^2)/(1 - x)^3.
E.g.f.: (8 + 30*x + 9*x^2)*exp(x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A081271(-n-2).
a(n) = 3*A095794(n+1) + 1.
a(n) = A034856(3*n+2) = A101881(6*n+2) = A165157(6*n+3) = A186349(6*n+3).
The inverse binomial transform yields 4, 15, 9, 0 (0 continued), therefore:
a(n) = 4*binomial(n,0) + 15*binomial(n,1) + 9*binomial(n,2).

A186346 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186347.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141

Offset: 1

Clark Kimberling, Feb 20 2011

nonn

See A186350 for a discussion of adjusted joint rank sequences.

			First, write
....8....16..24..32..40..48..56..64..72..80.. (8i)
1..4..9..16...25...36......49....64.......81 (squares)
Then replace each number by its rank, where ties are settled by ranking 8i before the square:
a=(3,5,7,9,11,12,14,15,17,..)=A186346
b=(1,2,4,6,8,10,13,16,19,...)=A186347.

Cf. A186350, A186347, A186348, A186349.

(* adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
 d=1/2; u=8; v=0; x=1; y=0;
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]];
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]];
Table[a[n],{n,1,120}]  (* A186346 *)
Table[b[n],{n,1,100}]  (* A186347 *)

a(n)=n+floor(sqrt(8n-1/2))=A186346(n).

b(n)=n+floor((n^2+1/2)/8)=A186347(n).

cp's OEIS Frontend

A186348 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186349.

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Mathematica

PARI

Formula

A262221 a(n) = 25*n*(n + 1)/2 + 1.

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Magma

Mathematica

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A186347 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186346.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A283394 a(n) = 3*n*(3*n + 7)/2 + 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Python

Sage

Formula

A186346 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=8i and g(j)=j^2. Complement of A186347.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A262221 a(n) = 25n(n + 1)/2 + 1.

A283394 a(n) = 3n(3*n + 7)/2 + 4.