A081589 -id:A081589 - cp's OEIS frontend

A081580 Pascal-(1,5,1) array.

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 61, 19, 1, 1, 25, 145, 145, 25, 1, 1, 31, 265, 595, 265, 31, 1, 1, 37, 421, 1585, 1585, 421, 37, 1, 1, 43, 613, 3331, 6145, 3331, 613, 43, 1, 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1, 1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016921, A081589, A081590. Coefficients of the row polynomials in the Newton basis are given by A013613.

			Square array begins as:
  1,  1,   1,    1,    1, ... A000012;
  1,  7,  13,   19,   25, ... A016921;
  1, 13,  61,  145,  265, ... A081589;
  1, 19, 145,  595, 1585, ... A081590;
  1, 25, 265, 1585, 6145, ...
The triangle begins as:
  1;
  1,  1;
  1,  7,    1;
  1, 13,   13,    1;
  1, 19,   61,   19,     1;
  1, 25,  145,  145,    25,     1;
  1, 31,  265,  595,   265,    31,     1;
  1, 37,  421, 1585,  1585,   421,    37,    1;
  1, 43,  613, 3331,  6145,  3331,   613,   43,    1;
  1, 49,  841, 6049, 17401, 17401,  6049,  841,   49,  1;
  1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1; - _Philippe Deléham_, Mar 15 2014

Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Cf. A002532, A016921, A081589, A081590.

A081580:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081580(n,k,5): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 6], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 6).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 5*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+5*x)^k/(1-x)^(k+1).
From Paul Barry, Aug 28 2008: (Start)
Number triangle T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*5^j.
Riordan array (1/(1-x), x*(1+5*x)/(1-x)). (End)
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 6). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(6*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 12*x + 36*x^2/2) = 1 + 13*x + 61*x^2/2! + 145*x^3/3! + 265*x^4/4! + 421*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n, k, 3) = A002532(n+1). - G. C. Greubel, May 26 2021

A214230 Sum of the eight nearest neighbors of n in a right triangular type-1 spiral with positive integers.

Original entry on oeis.org

53, 88, 78, 125, 85, 84, 125, 97, 108, 143, 223, 168, 158, 169, 201, 284, 208, 183, 179, 187, 210, 281, 226, 219, 227, 235, 261, 314, 430, 339, 311, 310, 318, 326, 346, 396, 515, 403, 360, 347, 355, 363, 371, 379, 411, 509, 427, 411, 419, 427, 435, 443, 451, 486, 557

Offset: 1

Alex Ratushnyak, Jul 08 2012

Right triangular type-1 spiral implements the sequence Up, Right-down, Left.
Right triangular type-2 spiral (A214251): Left, Up, Right-down.
Right triangular type-3 spiral (A214252): Right-down, Left, Up.
A140064 -- rightwards from 1: 3,14,34...
A064225 -- leftwards from 1: 8,24,49...
A117625 -- upwards from 1: 2,12,31...
A006137 -- downwards from 1: 6,20,43...
A038764 -- left-down from 1: 7,22,46...
A081267 -- left-up from 1: 9,26,52...
A081589 -- right-up from 1: 13, 61, 145...
9*x^2/2 - 19*x/2 + 6 -- right-down from 1: 5,18,40...

			Right triangular spiral begins:
56
55  57
54  29  58
53  28  30  59
52  27  11  31  60
51  26  10  12  32  61
50  25   9   2  13  33  62
49  24   8   1   3  14  34  63
48  23   7   6   5   4  15  35  64
47  22  21  20  19  18  17  16  36  65
46  45  44  43  42  41  40  39  38  37  66
78  77  76  75  74  73  72  71  70  69  68  67
The eight nearest neighbors of 3 are 1, 2, 13, 33, 14, 4, 5, 6. Their sum is a(3)=78.

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A214251, A214252.

Cf. A214226, A214231.

SIZE=29  # must be odd
grid = [0] * (SIZE*SIZE)
saveX = [0]* (SIZE*SIZE)
saveY = [0]* (SIZE*SIZE)
saveX[1] = saveY[1] = posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX,stepY,chkX,chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posY==0 or grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(0, -1,  1,  1)    # up
    if posY==0:
        break
    walk( 1, 1, -1,  0)    # right-down
    walk(-1, 0,  0, -1)    # left
for n in range(1,92):
    posX = saveX[n]
    posY = saveY[n]
    k = grid[(posY-1)*SIZE+posX] + grid[(posY+1)*SIZE+posX]
    k+= grid[(posY-1)*SIZE+posX-1] + grid[(posY-1)*SIZE+posX+1]
    k+= grid[(posY+1)*SIZE+posX-1] + grid[(posY+1)*SIZE+posX+1]
    k+= grid[posY*SIZE+posX-1] + grid[posY*SIZE+posX+1]
    print(k, end=', ')

A081272 Downward vertical of triangular spiral in A051682.

Original entry on oeis.org

1, 25, 85, 181, 313, 481, 685, 925, 1201, 1513, 1861, 2245, 2665, 3121, 3613, 4141, 4705, 5305, 5941, 6613, 7321, 8065, 8845, 9661, 10513, 11401, 12325, 13285, 14281, 15313, 16381, 17485, 18625, 19801, 21013, 22261, 23545, 24865, 26221, 27613, 29041, 30505

Offset: 0

Paul Barry, Mar 15 2003

Reflection of A081271 in the horizontal A051682.
Binomial transform of (1, 24, 36, 0, 0, 0, .....).
One of the six verticals of a triangular spiral which starts with 1 (see the link). Other verticals are A060544, A081589, A080855, A157889, A038764. - Yuriy Sibirmovsky, Sep 18 2016.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
Yuriy Sibirmovsky, Six verticals of the triangular spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682.

Cf. A062741, A060544, A081589, A080855, A157889, A038764.

Table[n^2 + (n + 1)^2, {n, 0, 300, 3}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 85}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
Table[n^2 + (n + 1)^2, {n, 0, 150, 3}] (* Vincenzo Librandi, Aug 07 2013 *)

x='x+O('x^99); Vec((1+22*x+13*x^2)/(1-x)^3) \\ Altug Alkan, Sep 18 2016

a(n) = C(n, 0) + 24*C(n, 1) + 36*C(n, 2).
a(n) = 18*n^2 + 6*n + 1.
G.f.: (1 + 22*x + 13*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(1 + 24*x + 18*x^2). - Stefano Spezia, Mar 07 2023

A081590 Fourth row of Pascal-(1,5,1) array A081580.

Original entry on oeis.org

1, 19, 145, 595, 1585, 3331, 6049, 9955, 15265, 22195, 30961, 41779, 54865, 70435, 88705, 109891, 134209, 161875, 193105, 228115, 267121, 310339, 357985, 410275, 467425, 529651, 597169, 670195, 748945, 833635, 924481, 1021699, 1125505, 1236115, 1353745, 1478611

Offset: 0

Paul Barry, Mar 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081580, A081589.

[1+36*n-54*n^2+36*n^3: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[36n^3-54n^2+36n+1,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,19,145,595},50] (* Harvey P. Dale, Sep 07 2011 *)

a(n) = 1 + 36*n - 54*n^2 + 36*n^3.
G.f.: (1+5*x)^3/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=19, a(2)=145, a(3)=595. - Harvey P. Dale, Sep 07 2011
E.g.f.: exp(x)*(1 + 18*x + 54*x^2 + 36*x^3). - Elmo R. Oliveira, Jun 06 2025

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1

Offset: 0

Peter Luschny, Mar 21 2023

A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).

			Array A(n, k) starts:
   [0] 1,  1,   1,    1,   1,   1,   1,    1, ...  A000012
   [1] 1,  3,   6,   10,  15,  21,  28,   36, ...  A000217
   [2] 1,  5,  13,   25,  41,  61,  85,  113, ...  A001844
   [3] 1,  7,  22,   46,  79, 121, 172,  232, ...  A038764
   [4] 1,  9,  33,   73, 129, 201, 289,  393, ...  A081585
   [5] 1, 11,  46,  106, 191, 301, 436,  596, ...  A081587
   [6] 1, 13,  61,  145, 265, 421, 613,  841, ...  A081589
   [7] 1, 15,  78,  190, 351, 561, 820, 1128, ...  A081591
   000012  | A028872 | A239325 |
       A005408    A100536   A069133
.
Triangle T(n, k) starts:
   [0] 1;
   [1] 1,  1;
   [2] 1,  3,   1;
   [3] 1,  6,   5,   1;
   [4] 1, 10,  13,   7,   1;
   [5] 1, 15,  25,  22,   9,   1;
   [6] 1, 21,  41,  46,  33,  11,   1;
   [7] 1, 28,  61,  79,  73,  46,  13,  1;
   [8] 1, 36,  85, 121, 129, 106,  61, 15,  1;
   [9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) =  1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) =  7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) =  1 = card('').

Rows: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591.
Columns: A000012, A005408, A028872, A100536, A239325, A069133.
Cf. A239592 (main diagonal), A239331 (transposed array).
Cf. A361045, A361043, A361521, A361731, A077028.

A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);

def A(m: int, steps: int) -> int:
    if m == 0: return 1
    size = m * steps
    cset = set()
    for a in range(0, size + 1, m):
        S = [str(int(i < a)) for i in range(size)]
        C = Combinations(S)
        cset.update("".join(i for i in c) for c in C)
    return len(cset)
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))

A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.

A155185 Primes in A155175.

Original entry on oeis.org

5, 13, 113, 1741, 5101, 8581, 9941, 21841, 26681, 47741, 82013, 481181, 501001, 1009621, 2356621, 2542513, 3279361, 3723721, 4277813, 7757861, 8124481, 13204661, 25311613, 30772013, 44170601, 48619661, 51521401, 52541501, 54236113, 60731221, 72902813

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Hypotenuse C (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes. p=1,q=2,a=3,b=4,c=5=prime,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[c],AppendTo[lst,c]]],{n,8!}];lst (* corrected by Ray Chandler, Feb 11 2020 *)

Sequence corrected by Ray Chandler, Feb 11 2020

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 13, 10, 1, 1, 9, 22, 25, 15, 1, 1, 11, 33, 46, 41, 21, 1, 1, 13, 46, 73, 79, 61, 28, 1, 1, 15, 61, 106, 129, 121, 85, 36, 1, 1, 17, 78, 145, 191, 201, 172, 113, 45, 1, 1, 19, 97, 190, 265, 301, 289, 232, 145, 55, 1, 1, 21

Offset: 0

Philippe Deléham, Mar 16 2014

			Square array begins:
n\k : 0......1......2......3......4......5......6......7......8......9
======================================================================
.0||  1......1......1......1......1......1......1......1......1......1
.1||  1......3......5......7......9.....11.....13.....15.....17.....19
.2||  1......6.....13.....22.....33.....46.....61.....78.....97....118
.3||  1.....10.....25.....46.....73....106....145....190....241....298
.4||  1.....15.....41.....79....129....191....265....351....449....559
.5||  1.....21.....61....121....201....301....421....561....721....901
.6||  1.....28.....85....172....289....436....613....820...1057...1324
.7||  1.....36....113....232....393....596....841...1128...1457...1828
.8||  1.....45....145....301....513....781...1105...1485...1921...2413
.9||  1.....55....181....379....649....991...1405...1891...2449...3079
10||  1.....66....221....466....801...1226...1741...2346...3041...3826
11||  1.....78....265....562....969...1486...2113...2850...3697...4654

Cf. Columns: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591, A081593.

Cf. Rows: A000012, A005408, A028872, A100536, A239325, A069133.

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k).
T(n,k) = 3*T(n,k-1) - 3*T(n,k-2) + T(n,k-3).
T(n,k) = (T(n,k-1) + T(n,k+1))/2 - A161680(n).
T(n,k) = (T(n-1,k) + T(n+1,k) - A000290(n))/2.

A155186 Primes in A155171.

Original entry on oeis.org

2, 7, 29, 101, 107, 197, 227, 457, 647, 829, 1549, 1627, 2221, 2309, 2347, 2521, 2677, 2801, 3181, 3299, 3529, 3541, 3557, 3739, 3769, 4231, 4549, 4871, 4987, 5651, 5827, 5881, 6037, 6079, 6637, 6827, 7517, 7639, 7937, 9787, 11621, 12041, 12329, 13009

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Numbers p (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[p],AppendTo[lst,p]]],{n,8!}];lst

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 3, 11, 71, 227, 491, 683, 1103, 1187, 2591, 3923, 4271, 4931, 6737, 7193, 7703, 8093, 8753, 8963, 9173, 9377, 10271, 13043, 13451, 13997, 15233, 15443, 15803, 15887, 17957, 18701, 19961, 20681, 21701, 22031, 22073, 24371, 24473, 24683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1, q=2(prime), a=3, b=4, c=5, s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150, A155186, A068231, A129517, A054574, A132281.

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[q],AppendTo[lst,q]]],{n,8!}];lst

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A081580 Pascal-(1,5,1) array.

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A214230 Sum of the eight nearest neighbors of n in a right triangular type-1 spiral with positive integers.

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A081272 Downward vertical of triangular spiral in A051682.

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A081590 Fourth row of Pascal-(1,5,1) array A081580.

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A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

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A155185 Primes in A155175.

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A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

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A155186 Primes in A155171.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.