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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A145215 a(n) is the minimal prime of the form 4k+1 for which s=A008784(n) is the minimal positive integer such that s*a(n)-floor(sqrt(s*a(n)))^2 is a square.

Original entry on oeis.org

5, 41, 353, 1237, 2749, 3037, 10369, 6569, 27253, 38561, 14897, 33289, 27917, 171629, 143513, 76081, 37649, 373273, 399181, 63029, 133157, 637601, 425197, 94261, 499321, 910853, 229849, 149837
Offset: 1

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Author

Vladimir Shevelev, Oct 05 2008

Keywords

Comments

See the conjecture in the comment at A145047. In addition, I conjecture that for every such s there exist infinitely many primes of the form 4k+1.

Crossrefs

Cf. A145016, A145022, A145023, A145047-A145050, A008487.

Programs

  • PARI
    f(s)=forprime(p=2,,if(p%4>1 || !issquare(s*p-sqrtint(s*p)^2),next);for(i=1,s-1,if(issquare(i*p-sqrtint(i*p)^2), next(2)));return(p))
S=select(n->if(n%2==0, if(n%4, n/=2, return(0))); n==1||vecmax(factor(n)[, 1]%4)==1, vector(150,i,i));
apply(f, S) \\ Charles R Greathouse IV, Feb 07 2013

Extensions

a(22) corrected by Charles R Greathouse IV, Feb 07 2013

A317023 Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals, where the sequence of row n is the expansion of (1-x^(n+1))/((1-x)^(n+1)).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 2, 0, 1, 5, 10, 9, 2, 0, 1, 6, 15, 20, 12, 2, 0, 1, 7, 21, 35, 34, 15, 2, 0, 1, 8, 28, 56, 70, 52, 18, 2, 0, 1, 9, 36, 84, 126, 125, 74, 21, 2, 0, 1, 10, 45, 120, 210, 252, 205, 100, 24, 2, 0, 1, 11, 55, 165, 330, 462, 461, 315, 130, 27, 2, 0, 1, 12, 66
Offset: 0

Views

Author

Werner Schulte, Jul 19 2018

Keywords

Comments

Conjecture: alternating row sums of the triangle give A106510 for n >= 0.

Examples

			The square array A(n,k) begins:
  n\k |  0  1  2   3    4    5    6     7     8     9     10
  ====+=====================================================
   0  |  1  0  0   0    0    0    0     0     0     0      0
   1  |  1  2  2   2    2    2    2     2     2     2      2
   2  |  1  3  6   9   12   15   18    21    24    27     30
   3  |  1  4 10  20   34   52   74   100   130   164    202
   4  |  1  5 15  35   70  125  205   315   460   645    875
   5  |  1  6 21  56  126  252  461   786  1266  1946   2877
   6  |  1  7 28  84  210  462  924  1715  2996  4977   7924
   7  |  1  8 36 120  330  792 1716  3432  6434 11432  19412
   8  |  1  9 45 165  495 1287 3003  6435 12870 24309  43749
   9  |  1 10 55 220  715 2002 5005 11440 24310 48620  92377
  10  |  1 11 66 286 1001 3003 8008 19448 43758 92378 184756
  etc.
The triangle T(n,k) begins:
  n\k |  0  1  2   3   4   5   6   7   8   9 10 11 12
  ====+==============================================
   0  |  1
   1  |  1  0
   2  |  1  2  0
   3  |  1  3  2   0
   4  |  1  4  6   2   0
   5  |  1  5 10   9   2   0
   6  |  1  6 15  20  12   2   0
   7  |  1  7 21  35  34  15   2   0
   8  |  1  8 28  56  70  52  18   2   0
   9  |  1  9 36  84 126 125  74  21   2   0
  10  |  1 10 45 120 210 252 205 100  24   2  0
  11  |  1 11 55 165 330 462 461 315 130  27  2  0
  12  |  1 12 66 220 495 792 924 786 460 164 30  2  0
  etc.

Crossrefs

Row sums of the triangle give A099036 for n >= 0.
Cf. A000984 (main diagonal), A000012 (column 0), A087156 (column 1).
Cf. A099036, A106510.
In the square array; row 0..12 are: A000007, A040000, A008486, A005893, A008487, A008488, A008489, A008490, A008491, A008492, A008493, A008494, A008495.
A173265 is based on the same square array, but is read by descending antidiagonals with special treatment of column 0.

Programs

  • GAP
    nmax:=15;; A:=List([0..nmax],n->List([0..nmax],k->Binomial(n+k,k)-Binomial(k-1,k-1-n)));;   b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->A[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Jul 20 2018
  • Mathematica
    Table[SeriesCoefficient[(1 - x^(# + 1))/((1 - x)^(# + 1)), {x, 0, k}] &[n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 20 2018 *)
  • PARI
    T(n,k) = binomial(n+k,k) - binomial(k-1,k-1-n); \\ Michel Marcus, Aug 07 2018

Formula

A(n,k) = binomial(n+k,k) - binomial(k-1,k-1-n) for n >= 0 and k >= 0 with binomial(i,j) = 0 if i < j or j < 0.
G.f.: Sum_{k>=0,n>=0} A(n,k)*x^k*y^n = ((1-x)^2)/((1-x-y)*(1-x-x*y)).
Seen as a triangle T(n,k) = A(n-k,k) = binomial(n,k)-binomial(k-1,2*k-1-n) for 0 <= k <= n with binomial(i,j) = 0 if i < j or j < 0.
Mirror image of the triangle equals A173265 except column 0.
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