A082611 -id:A082611 - cp's OEIS frontend

A080046 Multiplicative Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) * T(n-1,k).

1, 2, 2, 3, 4, 3, 4, 12, 12, 4, 5, 48, 144, 48, 5, 6, 240, 6912, 6912, 240, 6, 7, 1440, 1658880, 47775744, 1658880, 1440, 7, 8, 10080, 2388787200, 79254226206720, 79254226206720, 2388787200, 10080, 8, 9, 80640, 24078974976000

Offset: 1

Reinhard Zumkeller, Jan 21 2003

T(n,2) = A052849(n) for n>1.

Reinhard Zumkeller and Andre F. Labossiere, Rows n=1..14 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A082611.

Cf. A051597.

a080046 n k = a080046_tabl !! (n-1) !! (k-1)
a080046_row n = a080046_tabl !! (n-1)
a080046_tabl = iterate f [1] where
   f (x:xs) = [x + 1] ++ (zipWith (*) xs $ reverse xs) ++ [x + 1]
-- Reinhard Zumkeller, Oct 27 2013

Corrected by André F. Labossière, Sep 27 2004

A082607 a(0)=1; for n > 0, a(n) = least k not included earlier such that k*a(n-1) - 1 is a square.

Original entry on oeis.org

1, 2, 5, 10, 17, 26, 37, 50, 65, 34, 13, 25, 41, 61, 85, 113, 145, 122, 101, 82, 293, 634, 1105, 53, 109, 185, 74, 149, 250, 377, 205, 146, 97, 58, 29, 73, 137, 221, 181, 650, 541, 442, 353, 274, 953, 2042, 3541, 5450, 409, 173, 370, 289, 218, 157, 106, 337, 698

Offset: 0

Amarnath Murthy, Apr 28 2003

Conjecture: this is a permutation of A008784. - Robert Israel, Aug 25 2025

Robert Israel, Table of n, a(n) for n = 0..10000

Contained in A008784. Cf. A082608, A082609, A082610, A082611, A082612.

N:= 10000: # for terms before the first term > N
Cands:= select(t -> numtheory:-quadres(-1,t) = 1, [$2..N]): nc:= nops(Cands):
R:= 1: r:= 1:
do
  found:= false;
  for i from 1 to nc do
    if issqr(r*Cands[i]-1) then
       found:= true;
       r:= Cands[i];
       R:= R,r;
       Cands:= subsop(i=NULL,Cands);
       nc:= nc-1;
       break
    fi
  od;
  if not found then break fi
od:
R; # Robert Israel, Aug 25 2025

l = {1}; Do[k = 1; While[MemberQ[l, k] || !IntegerQ[Sqrt[k*Last[l]-1]], k++ ]; AppendTo[l, k], {n, 50}]; l (* Ryan Propper, Jun 13 2006 *)

a=[1];print1(1",");for(n=2,100,k=1;f=1;while(f,if(issquare(k*a[n-1]-1),f=0;for(i=1,n-1,if(a[i]==k,f=1)));k++);a=concat(a,k-1);print1(k-1",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

Corrected and extended by Ryan Propper, Jun 13 2006
Definition corrected by R. J. Mathar, Nov 12 2006
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A082608 Squares pertaining to A082607. a(n) = A082607(n)*A082607(n+1)- 1.

Original entry on oeis.org

1, 9, 49, 169, 441, 961, 1849, 3249, 2209, 441, 324, 1024, 2500, 5184, 9604, 16384, 17689, 12321, 8281, 24025, 185761, 700569, 58564, 5776, 20164, 13689, 11025, 37249, 94249, 77284, 29929, 14161, 5625, 1681, 2116, 10000, 30276, 40000, 117649

Offset: 0

Amarnath Murthy, Apr 28 2003

nonn

Cf. A082607, A082609, A082610, A082611, A082612.

Corrected and extended by R. J. Mathar, Nov 12 2006

A082610 Cubes pertaining to A082609. a(n) = A082609(n)*A082609(n+1)- 1.

Original entry on oeis.org

1, 27, 125, 512, 125000, 6708494456, 1978547007464, 9612511054106624, 59566081927816931517224, 911126473116338231047175614119973952, 57596301070572648147987649881578568811888391304423063406302778568

Offset: 0

Amarnath Murthy, Apr 28 2003

nonn

Cf. A082607, A082608, A082609, A082611, A082612.

Corrected and extended by David Wasserman, Oct 19 2004

cp's OEIS Frontend

A080046 Multiplicative Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) * T(n-1,k).

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A082607 a(0)=1; for n > 0, a(n) = least k not included earlier such that k*a(n-1) - 1 is a square.

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A082608 Squares pertaining to A082607. a(n) = A082607(n)*A082607(n+1)- 1.

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A082610 Cubes pertaining to A082609. a(n) = A082609(n)*A082609(n+1)- 1.

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