A213005 -id:A213005 - cp's OEIS frontend

A214961 a(0)=a(1)=1, a(n) = least k > a(n-1) such that k*a(n-2) is a triangular number.

1, 1, 3, 6, 7, 11, 13, 21, 25, 30, 49, 59, 97, 117, 193, 233, 385, 465, 492, 596, 983, 1191, 1965, 2381, 2516, 4761, 5031, 5761, 6290, 8466, 9795, 15470, 15867, 17403, 20559, 24170, 26945, 27192, 27755, 30130, 35235, 43537, 45100, 56805, 58717, 58739, 91000, 117477

Offset: 0

Alex Ratushnyak, Aug 03 2012

nonn

Robert Israel, Table of n, a(n) for n = 0..2100

Cf. A214914, A214915, A214916, A214963, A213005.

f:= proc(a,b)
  local s;
  s:= map(t -> rhs(op(t)), [msolve(x^2=1, 8*a)]);
  min(select(`>`, map(t -> (t^2-1)/(8*a), s), b))
end proc:
A[0]:= 1: A[1]:= 1:
for nn from 2 to 100 do
  A[nn]:= f(A[nn-2],A[nn-1])
od:
seq(A[i],i=0..100); # Robert Israel, Jun 17 2020

a[0]=a[1]=1;a[n_]:=a[n]=(k=a[n-1]+1;While[!IntegerQ@Sqrt[1+8*a[n-2]k],k++];k);Array[a,50,0] (* Giorgos Kalogeropoulos, May 21 2021 *)
lktn[{a_,b_}]:=Module[{k=b+1},While[!OddQ[Sqrt[8a k+1]],k++];{b,k}]; NestList[lktn,{1,1},50][[;;,1]] (* Harvey P. Dale, Sep 09 2023 *)

prpr = prev = 1
for n in range(1, 55):
    print(prpr, end=', ')
    b = k = 0
    while k<=prev:
        d = b*(b+1)//2
        k = 0
        if d%prpr==0:
            k = d // prpr
        b += 1
    prpr = prev
    prev = k

A212329 Expansion of x(5+x)/(1-7x+7*x^2-x^3).

Original entry on oeis.org

0, 5, 36, 217, 1272, 7421, 43260, 252145, 1469616, 8565557, 49923732, 290976841, 1695937320, 9884647085, 57611945196, 335787024097, 1957110199392, 11406874172261, 66484134834180, 387497934832825, 2258503474162776, 13163522910143837, 76722633986700252

Offset: 1

Kenneth J Ramsey, May 14 2012

Table of differences re Table A182441.
This is a sequence of differences between rows k and k+1 of table A182441. That is if A182441(k+1,0)-A182441(k,0) = 1, a(n) = A182441(k+1,n+1) - A182441(k,n+1) for n = 0 to 3. The remainder of the sequence is a continuation using the recursive formula D(n) = 6D(n-1)- D(n-2) + 6.
It appears that for n > 0, a(n) is divisible by A213005(n).
It appears that if p is a prime of the form 8*r +/- 1 then a(p-1) == 0 (mod p), and that if p is a prime of the form 8*r +/- 3 then a(p+1) == 0 (mod p).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A182441, A213005.

m = 12; n = 1; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m + 6; Sow[t];m = n; n = t;c++]][[2,1]]
CoefficientList[ Series[x (5 + x)/(1 - 7x + 7x^2 - x^3), {x, 0, 20}], x] (* or *)
LinearRecurrence[{7, -7, 1}, {0, 5, 36}, 21] (* Robert G. Wilson v, Jun 24 2014 *)

concat(0, Vec(x^2*(5+x)/((1-x)*(1-6*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 05 2016

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
From Colin Barker, Mar 05 2016: (Start)
a(n) = (-6+(5-3*sqrt(2))*(3+2*sqrt(2))^n + (3-2*sqrt(2))^n*(5+3*sqrt(2)))/4.
G.f.: x*(5+x) / ((1-x)*(1-6*x+x^2)).
(End)

A382245 Lexicographically earliest sequence of distinct nonnegative integers such that the product of two consecutive terms is always a triangular number (A000217).

Original entry on oeis.org

0, 1, 3, 2, 5, 9, 4, 7, 13, 6, 11, 21, 10, 12, 23, 45, 14, 15, 8, 17, 33, 16, 31, 61, 30, 26, 51, 25, 49, 24, 22, 43, 85, 42, 28, 55, 18, 35, 44, 87, 19, 37, 73, 36, 56, 111, 98, 195, 62, 69, 34, 39, 20, 41, 81, 40, 52, 103, 205, 66, 58, 115, 57, 29, 59, 117

Offset: 0

Rémy Sigrist, Mar 19 2025

nonn

This sequence has similarities with A026741: in both sequences, the product of two consecutive terms is always a triangular number; here all terms are distinct, there all products of two consecutive terms are distinct.

			The initial terms are:
  n   a(n)  a(n)*a(n+1)
  --  ----  ------------------
   0     0     0 = A000217(0)
   1     1     3 = A000217(2)
   2     3     6 = A000217(3)
   3     2    10 = A000217(4)
   4     5    45 = A000217(9)
   5     9    36 = A000217(8)
   6     4    28 = A000217(7)
   7     7    91 = A000217(13)
   8    13    78 = A000217(12)
   9     6    66 = A000217(11)
  10    11   231 = A000217(21)
  11    21   210 = A000217(20)
  12    10   120 = A000217(15)
  13    12   276 = A000217(23)
  14    23  1035 = A000217(45)
  15    45   630 = A000217(35)

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program

Cf. A000217, A026741, A077220, A213005, A382244.

PARI
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cp's OEIS Frontend

A214961 a(0)=a(1)=1, a(n) = least k > a(n-1) such that k*a(n-2) is a triangular number.

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A212329 Expansion of x(5+x)/(1-7x+7*x^2-x^3).

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A382245 Lexicographically earliest sequence of distinct nonnegative integers such that the product of two consecutive terms is always a triangular number (A000217).

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cp's OEIS Frontend

A214961 a(0)=a(1)=1, a(n) = least k > a(n-1) such that k*a(n-2) is a triangular number.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

A212329 Expansion of x*(5+x)/(1-7*x+7*x^2-x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382245 Lexicographically earliest sequence of distinct nonnegative integers such that the product of two consecutive terms is always a triangular number (A000217).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A212329 Expansion of x(5+x)/(1-7x+7*x^2-x^3).