A214838 -id:A214838 - cp's OEIS frontend

A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

1, 2, 8, 13, 47, 76, 274, 443, 1597, 2582, 9308, 15049, 54251, 87712, 316198, 511223, 1842937, 2979626, 10741424, 17366533, 62605607, 101219572, 364892218, 589950899, 2126747701, 3438485822, 12395593988, 20040964033, 72246816227, 116807298376

Offset: 0

Wolfdieter Lang, Nov 08 2002

-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n)= A077242(n).
The number a > 0 belongs to the sequence A077241, if a^2 belongs to the sequence A034856. - Alzhekeyev Ascar M, Apr 27 2012
Numbers k such that k^2 + 2 is a triangular number (see A214838). - Alex Ratushnyak, Mar 07 2013

			8*a(2)^2 + 17 = 8*8^2+17 = 529 = 23^2 = A077242(2)^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

I:=[1,2,8,13]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

LinearRecurrence[{0, 6, 0, -1}, {1, 2, 8, 13}, 30] (* Bruno Berselli, Mar 10 2013 *)
CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)

makelist(expand((-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8), n, 0, 30); /* Bruno Berselli, Mar 10 2013 */

a(2k) = A054488(k) and a(2k+1)= A077413(k) for k>=0.
G.f.: (1+x)*(1+x+x^2)/(1-6*x^2+x^4).
a(n) = (-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8. [Bruno Berselli, Mar 10 2013]

A328791 Triangular numbers of the form k^2 + 3.

Original entry on oeis.org

3, 28, 903, 30628, 1040403, 35343028, 1200622503, 40785822028, 1385517326403, 47066803275628, 1598885794044903, 54315050194251028, 1845112820810490003, 62679520857362409028, 2129258596329511416903, 72332112754346025765628, 2457162575051435364614403

Offset: 1

Jon E. Schoenfield, Oct 27 2019

nonn

There exist triangular numbers of the form k^2 + j for j=0 (A001110), j=1 (A164055), j=2 (A214838), and j=3 (this sequence), but not for j=4,7,8,13,16,18,... (A328792).

Cf. A001110, A164055, A214838, A328792.
Intersection of A000217 and A117950.
Cf. A276598 (the k's).

limit = 10**7 # rough limit for k
A000217 = set(k*(k+1)//2 for k in range(14*limit//10))
A117950 = set(k**2 + 3 for k in range(limit))
print(sorted(A000217 & A117950)) # Michael S. Branicky, Mar 28 2021

a(1) = 3, a(2) = 28; for n > 2, a(n) = 34*a(n-1) - a(n-2) - 46.

A328792 Numbers that are not the difference between any triangular number and the largest square that does not exceed it.

Original entry on oeis.org

4, 7, 8, 13, 16, 18, 22, 23, 25, 26, 31, 33, 34, 37, 38, 40, 43, 47, 48, 49, 52, 58, 59, 60, 61, 63, 64, 67, 68, 70, 73, 76, 79, 81, 83, 85, 86, 88, 92, 93, 94, 97, 98, 99, 102, 103, 106, 108, 112, 113, 114, 115, 118, 121, 123, 124, 125, 130, 133, 134, 138

Offset: 1

Jon E. Schoenfield, Oct 27 2019

nonn

			For any triangular number t, let f(t) = t - floor(sqrt(t))^2.
0 is not a term: for each term t in A001110, f(t) = 0.
1 is not a term: for each term t > 1 in A164055, f(t) = 1.
2 is not a term: for each term t in A214838, f(t) = 2.
3 is not a term: for each term t > 3 in A328791, f(t) = 3.
4 is a term, however: there exists no triangular number t such that f(t) = 4.

The complement of A230044.

Cf. A001110, A064784, A164055, A214838, A328791.

A305292 Numbers k such that k-1 is a square and k+1 is a triangular number.

Original entry on oeis.org

2, 5, 65, 170, 2210, 5777, 75077, 196250, 2550410, 6666725, 86638865, 226472402, 2943171002, 7693394945, 99981175205, 261348955730, 3396416785970, 8878171099877, 115378189547777, 301596468440090, 3919462027838450, 10245401755863185, 133146330756959525

Offset: 1

Bruno Berselli, Jun 11 2018

It is easy to prove that there are no numbers k such that k-1 is a triangular number and k+1 is a square.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. A000045, A000290, A077241, A214838.

LinearRecurrence[{1, 34, -34, -1, 1}, {2, 5, 65, 170, 2210}, 25]

Vec(x*(2 + 3*x - 8*x^2 + 3*x^3 + 2*x^4)/((1 - x)*(1 - 6*x + x^2)*(1 + 6*x + x^2)) + O(x^30)) \\ Colin Barker, Jun 14 2018

G.f.: x*(2 + 3*x - 8*x^2 + 3*x^3 + 2*x^4)/((1 - x)*(1 - 6*x + x^2)*(1 + 6*x + x^2)).
a(n) = a(-n-1) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5).
a(n) = 35*a(n-2) - 35*a(n-4) + a(n-6) = 34*a(n-2) - a(n-4) + 2.
a(n) = (-2 + (13*sqrt(2) + 7*(-1)^n)*(1 + sqrt(2))^(2*n+1) - (13*sqrt(2) - 7* (-1)^n)*(1 - sqrt(2))^(2*n+1))/32.
a(n) = A214838(n) - 1.
a(n) = A077241(n-1)^2 + 1.

cp's OEIS Frontend

A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

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A328791 Triangular numbers of the form k^2 + 3.

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A328792 Numbers that are not the difference between any triangular number and the largest square that does not exceed it.

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A305292 Numbers k such that k-1 is a square and k+1 is a triangular number.

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