A253446 -id:A253446 - cp's OEIS frontend

A253447 Indices of centered octagonal numbers (A016754) which are also centered heptagonal numbers (A069099).

1, 15, 435, 13021, 390181, 11692395, 350381655, 10499757241, 314642335561, 9428770309575, 282548466951675, 8467025238240661, 253728208680268141, 7603379235169803555, 227847648846413838495, 6827826086157245351281, 204606934935870946699921

Offset: 1

Colin Barker, Jan 01 2015

Also positive integers y in the solutions to 7*x^2 - 8*y^2 - 7*x + 8*y = 0, the corresponding values of x being A253446.

			15 is in the sequence because the 15th centered octagonal number is 841, which is also the 16th centered heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..678
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (31,-31,1).

Cf. A016754, A069099, A253446, A253514.

Vec(-x*(x^2-16*x+1)/((x-1)*(x^2-30*x+1)) + O(x^100))

a(n) = 31*a(n-1)-31*a(n-2)+a(n-3).
G.f.: -x*(x^2-16*x+1) / ((x-1)*(x^2-30*x+1)).
a(n) = (8+(4+sqrt(14))*(15+4*sqrt(14))^(-n)-(-4+sqrt(14))*(15+4*sqrt(14))^n)/16. - Colin Barker, Mar 03 2016

A253514 Centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 841, 755161, 678133681, 608963290321, 546848356574521, 491069215240629481, 440979608437728699361, 395999197307865131396641, 355606838202854450265484201, 319334544706965988473273415801, 286762065540017254794549261905041

Offset: 1

Colin Barker, Jan 03 2015

			841 is in the sequence because it is the 16th centered heptagonal number and the 15th centered octagonal number.

Colin Barker, Table of n, a(n) for n = 1..339
Index entries for linear recurrences with constant coefficients, signature (899,-899,1).

Cf. A001110, A016754, A069099, A157877, A157879, A253446, A253447.

Vec(-x*(x^2-58*x+1)/((x-1)*(x^2-898*x+1)) + O(x^100))

a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
G.f.: -x*(x^2-58*x+1) / ((x-1)*(x^2-898*x+1)).
From Peter Bala, Apr 15 2025; (Start)
a(n) = (1/64)*(-4 + sqrt(14))^2*(15 + 4*sqrt(14) + (449 + 120*sqrt(14))^n)^2 *(449 + 120*sqrt(14))^(-n).
a(-n) = a(n+1).
a(n) = (1/16) * (1 - T(2*n+1, -15)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. Cf. A001110.
a(n) = A157877(n)^2 = 1 + 7*A157879(n).
a(2) divides a(3*n+2); a(3) divides a(5*n+3); a(4) divides a(7*n+4); a(5) divides a(9*n+5). In general, a(k) divides a((2*k-1)*n + k). (End)

cp's OEIS Frontend

A253447 Indices of centered octagonal numbers (A016754) which are also centered heptagonal numbers (A069099).

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