A185438 -id:A185438 - cp's OEIS frontend

A188135 a(n) = 8n^2 + 2n + 1.

1, 11, 37, 79, 137, 211, 301, 407, 529, 667, 821, 991, 1177, 1379, 1597, 1831, 2081, 2347, 2629, 2927, 3241, 3571, 3917, 4279, 4657, 5051, 5461, 5887, 6329, 6787, 7261, 7751, 8257, 8779, 9317, 9871, 10441, 11027, 11629, 12247, 12881, 13531, 14197, 14879, 15577, 16291, 17021, 17767

Offset: 0

Paul Curtz, Mar 30 2011

Bisection of A193867. - Omar E. Pol, Aug 16 2011

Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A004770, A084849, A113770, A185438, A193867.

[1 + 2*n + 8*n^2: n in [0..50]]; // Vincenzo Librandi, Mar 30 2011

a(n)=8*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

First differences: a(n) - a(n-1) = 16*n - 6 = A113770(n) = 2*A004770(n).
Second differences: a(n) - 2*a(n-1) + a(n-2) = 16.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From R. J. Mathar, Apr 06 2011: (Start)
G.f.: -(1+x)*(7*x+1)/(x-1)^3.
a(n) = A084849(2*n). (End)
E.g.f.: exp(x)*(1 + 10*x + 8*x^2). - Elmo R. Oliveira, Oct 19 2024

a(41)-a(47) from Elmo R. Oliveira, Oct 19 2024

A193867 Odd central polygonal numbers.

Original entry on oeis.org

1, 7, 11, 29, 37, 67, 79, 121, 137, 191, 211, 277, 301, 379, 407, 497, 529, 631, 667, 781, 821, 947, 991, 1129, 1177, 1327, 1379, 1541, 1597, 1771, 1831, 2017, 2081, 2279, 2347, 2557, 2629, 2851, 2927, 3161, 3241, 3487, 3571, 3829, 3917, 4187, 4279

Offset: 1

Omar E. Pol, Aug 15 2011

Even triangular numbers plus 1.

Union of A188135 and A185438 without repetitions (A188135 is a bisection of this sequence. Another bisection is A185438 but without its initial term).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000124, A014494, A014601, A185438, A188135, A193868.

Select[Accumulate[Range[0,100]],EvenQ]+1 (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,7,11,29,37},50] (* Harvey P. Dale, Nov 29 2014 *)

Vec(-x*(x^2+1)*(x^2+6*x+1) / ((1+x)^2*(x-1)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016

a(n) = A000124(A014601(n-1)).
a(n) = 1 + A014494(n-1).
G.f.: -x*(x^2+1)*(x^2+6*x+1) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011
From Colin Barker, Jan 27 2016: (Start)
a(n) = (4*n^2+2*(-1)^n*n-4*n-(-1)^n+3)/2.
a(n) = 2*n^2-n+1 for n even.
a(n) = 2*n^2-3*n+2 for n odd. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(2*cosh(sqrt(7)*Pi/4) - sqrt(2))). - Amiram Eldar, May 11 2025

A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.

Original entry on oeis.org

1, 1, 2, 7, 11, 2, 11, 29, 37, 23, 28, 67, 79, 23, 53, 121, 137, 77, 86, 191, 211, 29, 127, 277, 301, 163, 176, 379, 407, 109, 233, 497, 529, 281, 298, 631, 667, 88, 371, 781, 821, 431, 452, 947, 991, 259, 541, 1129, 1177, 613, 638

Offset: 0

Jean-François Alcover and Paul Curtz, Apr 29 2014

The subsequence 1, 23, 77, 163, 281, 431, 613, 827, ..., with indices congruent to 1 mod 8, is 16n^2+6n+1, that is, A000124(8n+1)/2 or A014206(8n+1)/4. Its second differences are constant: (16n^2+6n+1)'' = 32.

The sequence A014206/A241807 is integral and consists of the 16-periodic sequence (2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2, ...).

			1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, 37/495, 23/330, ...

Cf. A000124, A014206, A241269, A188135, A054552, A185438.

Table[(n^2+n+2)/((n+1)*(n+2)*(n+3)) // Numerator, {n, 0, 50}]

a(n) = A014206(n)/period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2 (conjectured).
a(4k)     = 8*k^2 +2*k +1,
a(4k+2)   = 4*k^2 +5*k +2,
a(4k+3)   = 8*k^2 +14*k +7,
a(8k+1)   = 16*k^2 +6*k +1,
a(16k+5)  = 16*k^2 +11*k +2,
a(16k+13) = 32*k^2 + 54*k +23.

cp's OEIS Frontend

A188135 a(n) = 8n^2 + 2n + 1.

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A193867 Odd central polygonal numbers.

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A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.

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cp's OEIS Frontend

A188135 a(n) = 8*n^2 + 2*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

PARI

Formula

Extensions

A193867 Odd central polygonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A241807 Numerators of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)) as defined in A241269.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A188135 a(n) = 8n^2 + 2n + 1.

A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.