A028557 -id:A028557 - cp's OEIS frontend

A347672 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of Baxter matrices of size n X k.

1, 1, 1, 1, 6, 1, 1, 14, 14, 1, 1, 24, 69, 24, 1, 1, 36, 203, 203, 36, 1, 1, 50, 463, 972, 463, 50, 1, 1, 66, 903, 3324, 3324, 903, 66, 1, 1, 84, 1585, 9074, 16355, 9074, 1585, 84, 1, 1, 104, 2579, 21168, 61267, 61267, 21168, 2579, 104, 1, 1, 126, 3963, 44028, 188153, 306352, 188153, 44028, 3963, 126, 1

Offset: 1

N. J. A. Sloane, Sep 10 2021

			The array begins:
1,1,1,1,1,1,1, ...
1,6,14,24,36,50,66, ...
1,14,69,203,463,903,1585, ...
1,24,203,972,3324,9074,21168, ...
1,36,463,3324,16355,61267,188153, ...
1,50,903,9074,61267,306352,1219598, ...
1,66,1585,21168,188153,1219598,6175181, ...
...
The first few antidiagonals are:
1,
1,1,
1,6,1,
1,14,14,1,
1,24,69,24,1,
1,36,203,203,36,1,
1,50,463,972,463,50,1,
1,66,903,3324,3324,903,66,1,
1,84,1585,9074,16355,9074,1585,84,1,
...

Michael S. Branicky, Table of n, a(n) for n = 1..96
Don Knuth, Baxter matrices, Preprint, Sep 05 2021.
George Spahn, Counting Baxter Matrices, arXiv:2110.09688 [math.CO], 2021.

Row 2 is A028557, row 3 is A347673, main diagonal is A347674.

Cf. A347675, A347676.

a(25) corrected by and a(46)-a(66) from Michael S. Branicky, Sep 14 2021

A020739 a(n) = 2*n + 6.

Original entry on oeis.org

6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138

Offset: 0

David W. Wilson

Pisot sequence T(6,8).

Trivial case of a Pisot sequence satisfying a simple linear recurrence. Here, since round((2*n+2)^2/(2*n)^2) = 2*n + round((n+1)/n^2) = 2*n for n > 2, a(n) is even and a(n) = a(n-1) + 2. - Ralf Stephan, Sep 03 2013

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A005843. See A008776 for definitions of Pisot sequences.

Cf. A009056, A028557.

2*Range[0,70]+6 (* or *) Range[6,138,2] (* Harvey P. Dale, Apr 24 2017 *)

a(n) = 2*a(n-1) - a(n-2).
From Elmo R. Oliveira, Oct 30 2024: (Start)
G.f.: 2*(3 - 2*x)/(1 - x)^2.
E.g.f.: 2*exp(x)*(3 + x).
a(n) = 2*A009056(n+1) = A028557(n+1) - A028557(n). (End)

Better name from Ralf Stephan, Sep 03 2013

A116351 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m+9.

Original entry on oeis.org

8446, 8811, 69125298546226023972, 69855225553525294045, 74604750601020544520, 75334677608319814593, 92496418993920707747, 93226346001219977820, 97975871048715228295, 98705798056014498368

Offset: 1

Giovanni Resta, Feb 06 2006

			8811 * 8816 = 7767//7776, where // denotes concatenation.

Cf. A028557, A116180, A116338, A116350, A116352.

A132770 a(n) = n*(n + 28).

Original entry on oeis.org

0, 29, 60, 93, 128, 165, 204, 245, 288, 333, 380, 429, 480, 533, 588, 645, 704, 765, 828, 893, 960, 1029, 1100, 1173, 1248, 1325, 1404, 1485, 1568, 1653, 1740, 1829, 1920, 2013, 2108, 2205, 2304, 2405, 2508, 2613, 2720, 2829, 2940, 3053, 3168, 3285, 3404, 3525

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769.

#(#+28)&/@Range[0,50] (* or *) LinearRecurrence[{3,-3,1},{0,29,60},50] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=n*(n+28) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+28) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 27, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(28)/28 = A001008(28)/A102928(28) = 315404588903/2248776129600, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7751493599/321253732800. (End)
G.f.: x*(29 - 27*x)/(1-x)^3. - Harvey P. Dale, Aug 03 2021
E.g.f.: x*(29 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A132771 a(n) = n*(n + 29).

Original entry on oeis.org

0, 30, 62, 96, 132, 170, 210, 252, 296, 342, 390, 440, 492, 546, 602, 660, 720, 782, 846, 912, 980, 1050, 1122, 1196, 1272, 1350, 1430, 1512, 1596, 1682, 1770, 1860, 1952, 2046, 2142, 2240, 2340, 2442, 2546, 2652, 2760, 2870, 2982, 3096, 3212, 3330, 3450, 3572

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.

Table[n(n+29), {n,0,50}] (* Bruno Berselli, Apr 03 2015 *)
LinearRecurrence[{3,-3,1},{0,30,62},50] (* Harvey P. Dale, Oct 18 2024 *)

a(n)=n*(n+29) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+29) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 28 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(29)/29 = A001008(29)/A102928(29) = 9227046511387/67543597321200, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/29 - 236266661971/9649085331600. (End)
From G. C. Greubel, Mar 13 2022: (Start)
G.f.: 2*(15*x - 14*x^2)/(1-x)^3.
E.g.f.: x*(30 + x)*exp(x). (End)

A132772 a(n) = n*(n + 30).

Original entry on oeis.org

0, 31, 64, 99, 136, 175, 216, 259, 304, 351, 400, 451, 504, 559, 616, 675, 736, 799, 864, 931, 1000, 1071, 1144, 1219, 1296, 1375, 1456, 1539, 1624, 1711, 1800, 1891, 1984, 2079, 2176, 2275, 2376, 2479, 2584, 2691, 2800, 2911, 3024, 3139, 3256, 3375, 3496, 3619

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A067079, A098849, A098850, A098603, A098847, A098848, A102928, A120071.
Cf. A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767.
Cf. A132768, A132769, A132770, A132771.

Table[n(n+30),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,31,64},50] (* Harvey P. Dale, Mar 06 2015 *)

a(n)=n*(n+30) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+30) for n in (0..50)] # G. C. Greubel, Mar 13 2022

G.f.: x*(31-29*x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*n + a(n-1) + 29 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=31, a(2)=64, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 06 2015
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(30)/30 = A001008(30)/A102928(30) = 9304682830147/69872686884000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 225175759291/9981812412000. (End)
E.g.f.: x*(31 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A249013 a(n) = floor( (n-1) * (n+4) / 10 ).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 6, 8, 10, 12, 15, 17, 20, 23, 26, 30, 33, 37, 41, 45, 50, 54, 59, 64, 69, 75, 80, 86, 92, 98, 105, 111, 118, 125, 132, 140, 147, 155, 163, 171, 180, 188, 197, 206, 215, 225, 234, 244, 254, 264, 275, 285, 296, 307, 318, 330, 341, 353, 365

Offset: 1

Michael Somos, Oct 19 2014

A028557(n) without the least significant digit. - R. J. Mathar, Aug 11 2021

			G.f. = x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + 12*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A249020, A028557.

[Floor((n-1)*(n+4)/10): n in [1..60]]; // Vincenzo Librandi, Jan 10 2015

a[ n_] := Quotient[ (n - 1) (n + 4), 10];
LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 0, 1, 2, 3, 5, 6}, 60] (* or *) CoefficientList[Series[x^2 (1 + x^3 - x^4) / ((1 - x)^2 (1 - x^5)), {x, 0, 60}], x] (* Vincenzo Librandi, Jan 10 2015 *)

PARI
```
{a(n) = (n-1) * (n+4) \ 10};
```

{a(n) = if( n<-1, n = -3-n); -(n<1) + polcoeff( x^3*(1 + x^3 - x^4) / ((1 - x)^2 * (1 - x^5)) + x * O(x^n), n)};

G.f.: x^3 * (1 + x^3 - x^4) / ((1 - x)^2 * (1 - x^5)) = x^3*(1+x^3-x^4)/ ( (1-x)^3*(1+x+x^2+x^3+x^4)).
a(n) = a(-3-n) for all n in Z.
a(n) = a(n-5) + n-1 for all n in Z.
a(n) + a(n+4) = min( a(n+1) + a(n+3), a(n+2) + a(n+2) ) + 1 for all n in Z.
A249020(n) = a(n+1) + 1 for all n in Z. - Michael Somos, Jan 09 2015

A104675 a(n) = C(n+1,n) * C(n+6,1).

Original entry on oeis.org

6, 14, 24, 36, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646

Offset: 0

Zerinvary Lajos, Apr 22 2005

			If n=0 then C(0+1,0+0) * C(0+6,1) = C(1,0) * C(6,1) = 1*6 = 6.
If n=5 then C(5+1,5+0) * C(5+6,1) = C(6,5) * C(11,1) = 6*11 = 66.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A028557, A062190.

[(n+1)*(n+6): n in [0..50]]; // G. C. Greubel, Mar 01 2025

Table[Binomial[n + 1, n] Binomial[n + 6, 1], {n, 0, 48}] (* or *)
CoefficientList[Series[2 (3 - 2 x)/(1 - x)^3, {x, 0, 49}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {6, 14, 24}, 49] (* Michael De Vlieger, Apr 06 2017 *)

Vec(2*(3 - 2*x) / (1 - x)^3 + O(x^80)) \\ Colin Barker, Apr 06 2017

a(n)=(n+6)*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

from sympy import binomial
def a(n): return binomial(n + 1, n) * binomial(n + 6, 1) # Indranil Ghosh, Apr 06 2017

a(n) = (n+1)*(n+6) = A028557(n+1). - R. J. Mathar, May 19 2008
a(n) = 2*n + a(n-1) + 6 (with a(0)=6). Vincenzo Librandi, Nov 13 2010
From Colin Barker, Apr 06 2017: (Start)
G.f.: 2*(3 - 2*x) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
E.g.f.: exp(x)*(x^2 + 8x + 6). - Indranil Ghosh, Apr 06 2017
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 137/300.
Sum_{n>=0} (-1)^n/a(n) = 2*log(2)/5 - 47/300. (End)

A028559 Palindromes of the form k*(k+5).

Original entry on oeis.org

0, 6, 66, 414, 696, 41814, 42224, 666666, 4282824, 4754574, 4881884, 416343614, 630939036, 4159669514, 6817557186, 42877777824, 4163250523614, 4783601063874, 4986733376894, 47431877813474, 6333914444193336, 44653247574235644, 62141509790514126

Offset: 1

Patrick De Geest

Lior Manor, Table of n, a(n) for n = 1..33 (terms 1..27 from Michael S. Branicky)
Patrick De Geest, Palindromic Quasipronics of the form n(n+x)

Cf. A028557, A028558.

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen():
    for k in count(0):
        if ispal(k*(k+5)):
            yield k*(k+5)
print(list(islice(agen(), 20))) # Michael S. Branicky, Jan 30 2022

a(n) = A028558(n) * (A028558(n) + 5). - Michael S. Branicky, Jan 30 2022

a(21) and beyond from Michael S. Branicky, Jan 30 2022

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

cp's OEIS Frontend

A347672 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of Baxter matrices of size n X k.

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A020739 a(n) = 2*n + 6.

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Mathematica

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Extensions

A116351 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m+9.

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A132770 a(n) = n*(n + 28).

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Mathematica

PARI

Sage

Formula

A132771 a(n) = n*(n + 29).

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Mathematica

PARI

Sage

Formula

A132772 a(n) = n*(n + 30).

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Programs

Mathematica

PARI

Sage

Formula

A249013 a(n) = floor( (n-1) * (n+4) / 10 ).

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Magma

Mathematica

PARI

PARI

Formula

A104675 a(n) = C(n+1,n) * C(n+6,1).

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