A168234 -id:A168234 - cp's OEIS frontend

A171710 Union of A168234 and A171219, sorted.

3, 3, 5, 5, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89

Offset: 1

Paul Curtz, Dec 16 2009

Consider a table T(n,k) similar to A168142 = {2,1}, {8,7,6,...,2,1}, {18,17,...,2,1},... that repeats each row. Thus T(n,k) = {2,1}, {2,1}, {8,7,6,...,2,1}, {8,7,6,...,2,1}, {18,17,...,2,1}, etc. The rows of T(n,k) decrement from 2*ceiling(n/2)^2 to 1. Then we can construct the table of atomic numbers in the Janet periodic table A138509(n) = T(n,k) + a(n), with k=2*ceiling(n/2)^2 - 1 down to 1 by step -1.

Michael De Vlieger, Table of n, a(n) for n = 1..10680 (first 39 rows)

Cf. A138509 (Janet periodic table, rows n > 1 end in the repeated numbers in this sequence), A168234 (odd rows), A168380 (repeated numbers k - 1), A171219 (even rows), A172002 (smallest values of rows n > 1 are the repeated numbers in this sequence).

Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 7}, {k, 2 Ceiling[n/2]^2}] // Flatten (* Michael De Vlieger, Jul 20 2016 *)

May be regarded as an irregular triangle read by rows, defined by T(n,k) = A168380(n) + 1, with 1 <= k <= ceiling(n/2)^2. - Michael De Vlieger, Jul 20 2016

A168388 First number in the n-th row of A172002.

Original entry on oeis.org

1, 3, 5, 13, 21, 39, 57, 89, 121, 171, 221, 293, 365, 463, 561, 689, 817, 979, 1141, 1341, 1541, 1783, 2025, 2313, 2601, 2939, 3277, 3669, 4061, 4511, 4961, 5473, 5985, 6563, 7141, 7789, 8437, 9159, 9881, 10681, 11481, 12363, 13245, 14213, 15181, 16239, 17297

Offset: 1

Paul Curtz, Nov 24 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Cf. A168234.

[(12+n+3*(-1)^n*n+2*n^3)/12: n in [1..60]]; // Vincenzo Librandi, Jul 20 2016

LinearRecurrence[{2,1,-4,1,2,-1},{1,3,5,13,21,39},50] (* Harvey P. Dale, Nov 29 2014 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 0, 46}] (* Michael De Vlieger, Jul 19 2016, after Vincenzo Librandi at A168380 *)

a(2*n+1) = A166464(n)        a(2*n) = A166911(n).
a(n+1) - a(n) = A093907(n-1), n>1.
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).
G.f.: x*(1 - x^2 + 2*x)*(1 - x + x^2 + x^3)/( (1+x)^2 * (x-1)^4).
a(n+1) = A168380(n)+1.
From G. C. Greubel, Jul 19 2016: (Start)
a(n) = (12 + n + 3*(-1)^n*n + 2*n^3)/12.
E.g.f.: (1/12)*( -3*x - 12*exp(x) + (12 + 3*x + 6*x^2 + 2*x^3)*exp(2*x) )*exp(-x). (End)

Edited and extended by R. J. Mathar, Mar 25 2010

A168574 a(n) = (4n + 3)(1 + 2*n^2)/3.

Original entry on oeis.org

1, 7, 33, 95, 209, 391, 657, 1023, 1505, 2119, 2881, 3807, 4913, 6215, 7729, 9471, 11457, 13703, 16225, 19039, 22161, 25607, 29393, 33535, 38049, 42951, 48257, 53983, 60145, 66759, 73841, 81407, 89473, 98055, 107169, 116831, 127057, 137863, 149265, 161279

Offset: 0

Paul Curtz, Nov 30 2009

Binomial transform of quasi-finite sequence 1, 6, 20, 16, 0, 0, ... (0 continued).

a(n+1) is the sum of the first and last number at the bottom (2nd row) of each block in A172002, 3+4, 13+20, 39+56, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A168547, A168234.

[(4*n+3)*(1+2*n^2)/3 : n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Table[ (4*n+3)*(1+2*n^2)/3 , {n,0,25}] (* G. C. Greubel, Jul 26 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,7,33,95},40] (* Harvey P. Dale, May 16 2019 *)

a(n)=(4*n+3)*(1+2*n^2)/3 \\ Charles R Greathouse IV, Jul 26 2016

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 16.
a(n) = A168582(2*n+1) .
a(n+1) = A166911(n) + A002492(n+1).
G.f.: (1 + 3*x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (1/3)*(3 + 18*x + 30*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Jul 26 2016

Edited and extended by R. J. Mathar, Mar 25 2010

A171219 A138101(n)+A168142(n).

Original entry on oeis.org

5, 5, 21, 21, 21, 21, 21, 21, 21, 21, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 1

Paul Curtz, Dec 05 2009

This here basically repeats entries A168388(2k+1) 2*k^2 times for k=1,2,....

The construction is similar to A168234.

Comments tightened - R. J. Mathar, Nov 24 2010

cp's OEIS Frontend

A171710 Union of A168234 and A171219, sorted.

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A168388 First number in the n-th row of A172002.

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A168574 a(n) = (4*n + 3)*(1 + 2*n^2)/3.

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A171219 A138101(n)+A168142(n).

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A168574 a(n) = (4n + 3)(1 + 2*n^2)/3.