A080919 -id:A080919 - cp's OEIS frontend

A080900 a(1)=1; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

1, 6, 11, 16, 21, 19, 24, 29, 34, 39, 37, 42, 47, 52, 57, 55, 60, 65, 63, 68, 66, 71, 76, 74, 79, 84, 89, 94, 92, 97, 102, 107, 112, 110, 115, 120, 118, 123, 121, 126, 131, 129, 134, 139, 144, 149, 147, 152, 157, 162, 167, 165, 170, 175, 173, 178, 176

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Apr 01 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000 (terms 1..1000 from Ivan Neretin)
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 2.

Cf. A080912, A080913, A080904, A080914, A080578, A080919, A080922, A080927.

Cf. A080901 (starting value = 2), A080905 (run lengths of first differences).

Fold[Append[#1, #1[[-1]] + If[MemberQ[#1, #2], -2, 5]] &, {1}, Range[2, 57]] (* Ivan Neretin, Mar 03 2016 *)

up_to = 1001;
A080900list(up_to_n) = { my(xs=Map(), v=vector(up_to_n)); mapput(xs,1,1); v[1] = 1; for(n=2,up_to_n, v[n] = v[n-1]+if(mapisdefined(xs,n), -2, +5); mapput(xs,v[n],n)); (v); };
v080900 = A080900list(up_to);
A080900(n) = v080900[n]; \\ Antti Karttunen, Jan 22 2020

Perhaps this is asymptotic to c_0*n*(1 + c_1/log n + ...), with c_0 near 2 ?

A080853 Square array of generalized polygonal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 7, 1, 1, 5, 16, 19, 11, 1, 1, 6, 25, 37, 33, 16, 1, 1, 7, 36, 61, 67, 51, 22, 1, 1, 8, 49, 91, 113, 106, 73, 29, 1, 1, 9, 64, 127, 171, 181, 154, 99, 37, 1, 1, 10, 81, 169, 241, 276, 265, 211, 129, 46, 1, 1, 11, 100, 217, 323, 391, 406, 365, 277

Offset: 0

Paul Barry, Feb 23 2003

			Rows begin with n>=0, k>=0
1 1 1 1 1 ...
1 2 4 7 11 ...
1 3 9 19 33 ...
1 4 16 37 67 ...
1 5 25 61 113 ...

Rows include A000124, A058331, A080855, A080856, A080857, A080919.

Columns include A000290, A003215, A003215, A080859, A080860, A080861.

A080853 := proc(n,k)
    binomial(k,0)+n*binomial(k,1)+n^2*binomial(k,2) ;
end proc:
seq( seq(A080853(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

T(n, k)=C(k, 0)+C(k, 1)n+C(k, 2)n^2=(n^2*k^2-(n^2-2n)*k+2)/2 =(k(k-1)*n^2+2k*n+2)/2
Row n has g.f. (1+(n-2)x+(n^2-n+1)x^2)/(1-x)^3.
Column k has g.f. (C(k-1, 0)+(C(k+1, 2)-2)*x+C(k-1, 2)*x^2)/(1-x)^3.
Diagonals are given by (n^4+(2k-1)*n^3+((k-1)^2+1)*n^2+(1-(k-1)^2)*n+2)/2.
Antidiagonal sums are 1, 2, 4, 9, 22, 53, 119,... = (d+1)*(2*d^4-7*d^3+27*d^2-22*d+120)/120 = sum_{k=0..d} T(d-k,k), first differences in  A116701, d>=0. - R. J. Mathar, Oct 01 2021

A080914 A080900 sorted and duplicates removed.

Original entry on oeis.org

1, 6, 11, 16, 19, 21, 24, 29, 34, 37, 39, 42, 47, 52, 55, 57, 60, 63, 65, 66, 68, 71, 74, 76, 79, 84, 89, 92, 94, 97, 102, 107, 110, 112, 115, 118, 120, 121, 123, 126, 129, 131, 134, 139, 144, 147, 149, 152, 157, 162, 165, 167, 170, 173, 175, 176, 178

Offset: 1

N. J. A. Sloane, Apr 02 2003

nonn

Cf. A080900, A080919.

sort -n -u A080900

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A080900 a(1)=1; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

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A080853 Square array of generalized polygonal numbers, read by antidiagonals.

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A080914 A080900 sorted and duplicates removed.

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